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University  of  California  •  Berkeley 

The  Theodore  P.  Hill  Collection 
Early  American  Mathematics  Books 


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ELEMENTS 


OF 


ANALYTIC  GEOMETRY. 


BY 


ARTHUR  SHERBURNE  HARDY,  Ph.D., 

Professor  op  Mathematics  in  Dartmouth 
College. 


-»o>»ioo- 


BOSTON,  U.S.A.: 
PUBLISHED   BY  GIXN  &  COMPANY. 

■  1889. 


Copyright,  1888, 
Bt  ARTHUR  SHERBURNE  HARDY. 


All  Rights  Reserved. 


Typography  by  J.  8.  Cusuing  &  Co.,  Boston,  U.S.A. 


Presswork  by  Ginn  &  Co.,  Boston,  U.S.A. 


PEEFACE. 


Although  writing  a  text-book  for  the  use  of  beginners 
following  a  short  course,  the  teudencj'  of  an  author  is  to 
sacrifice  the  practical  value  of  the  treatise  to  completeness, 
generalization,  and  scientific  presentation.  I  have  endeavored 
to  avoid  this  error,  which  renders  many  works  unsuitable  for 
the  class-room,  however  valuable  they  may  be  for  reference, 
and  yet  to  encourage  the  habit  of  generalization.  To  this 
end  I  have  attempted  to  shun  the  difficulties  involved  in 
introducing  the  beginner  to  the  conies,  before  he  is  familiar 
with  their  forms,  through  the  discussion  of  the  general  equa- 
tion ;  and  at  the  same  time  to  secure  to  him  the  advantages 
of  a  general  analysis  of  the  equation  of  the  second  degree. 
The  teacher  will  observe  the  same  effort  to  cultivate  the 
power  of  general  reasoning,  which  it  is  one  of  the  objects 
of  Analytic  Geometry  to  promote,  in  the  preliminary  con- 
struction of  loci,  a  process  too  often  left  in  the  form  of  a 
merely  mechanical  construction  of  points  by  substitution  in 
the  equation.  In  passing  from  Geometry  to  Analytic 
Geometry,  the  student  should  see  that,  while  the  field  of 
operations  is  extended,  the  subject  matter  is  essentially  the 
same ;  and  that  what  is  fundamentally  new  is  the  method, 
the  lines  and  surfaces  of  Geometry  being  replaced  by  their 
equations.     His  chief  difficulties  are  : 


IV  PREFACE. 

First.  A  thorough  uuderstanding  of  the  device  by  which 
this  substitution  is  effected  ;  hence  considerable  attention  has 
been  paid  to  this  simple  matter. 

Second.  The  acquisition  of  an  independent  use  of  the 
new  method  as  an  instrument  of  researcli ;  hence  the  inser- 
tion of  problems  illustrative  of  tlie  analytic,  as  distinguished 
from  the  geometric,  method  of  proof.  The  function  of 
numerical  examples  —  tl)at  is,  of  examples  consisting  of  a 
mere  substitution  of  numerical  values  for  the  general  con- 
stants—  is  simply  that  of  testing  the  student's  knowledge  of 
the  nomenclature.  The  real  example  in  Analytic  Geometry 
is  the  application  of  the  method  to  the  discovery  of  geomet- 
rical properties  and  forms. 

The  polar  system  has  been  freely  used.  It  is  not  briefly 
explained  and  subsequently  abandoned  without  application ; 
nor  is  it  applied  redundantly  to  what  has  been  already 
treated  by  the  rectilinear  system.  It  is  used  as  one  of  two 
methods,  each  of  which  has  its  advantages,  the  selection  of 
one  or  the  other  in  any  given  case  being  governed  by  its 
adaptability  to  the  demonstration  or  problem  in  hand. 

The  time  allotted  to  the  courses  in  Analytic  Geometry  for 
which  it  is  hoped  this  treatise  will  be  found  adapted  has 
determined  the  exclusion  of  certain  topics,  and  has  limited 
the  chapters  on  Solid  Geometry  to  the  elements  necessary  to 
the  student  in  the  subsequent  study  of  Analytic  Mechanics. 

■  ARTHUR    SHERBURNE   HARDY. 
Hanover,  N.II.,  Oct.  6,  1888. 


CONTENTS. 


Part  I.  —  Plane  Analytic  Geometry. 


CHAPTER  I.  —  COORDINATE  SYSTEMS. 
Section  I.  —  The  Point. 
The  Rectilinear  System. 

ART.  PAGE. 

1.  Position  of  a  point  in  a  plane 1 

2.  Definitions  .  , 2 

3.  Construction  of  a  point 2 

4.  Definitions 2 

5.  Equations  of  a  point.     Examples 3 

6.  Division  of  a  line.     Examples 4 

7.  Distance  between  two  points.     Examples 5 

The  Polar  System. 

8.  Position  of  a  point  in  a  plane 6 

9.  Signs  of  the  polar  coordinates 7 

10.  Construction  of  a  point 7 

11.  Equations  of  a  point 7 

12.  Definitions.     Examples 8 

13.  Distance  between  two  points.     Examples 8 

Section  TT.  —  The  Line. 
The  Rectilinear  System. 

14.  Loci,  and  their  equations 10 

15.  Distinctions  between  Analj-tic  Geometry,  Geometry,  and  Algebra  11 

16.  Quantities  of  Analytic  Geometry 12 

17.  Construction  of  loci.     Examples 13 


VI  CONTENTS. 

The  Polar  System. 

ART.  PAGE. 

18.  Polar  equations  of  loci 21 

19.  Construction  of  polar  equations 22 

20.  General  notation 2i 

Section  III.  —  Relation  between  the  Rectilinear  and  Polar 

Systems. 

21.  Transformation  of  coordinates 25 

Eectilinear  Transformations. 

22.  Formulas  for  passing  from  ;iny  rectilinear  system  to  any  other  . .       27 

Polar  Transformations. 

23.  Formulas  for  passing  from  any  rectilinear  to  any  polar  system  . .       30 

24.  Formulae  for  passing  from  any  polar  to  any  rectilinear  system  . .       31 


CHAPTER    II,   —  EQUATION     OP     THE    FIRST    DEGREE.  THE 

STRAIGHT   LINE. 

Section  IV. — The  Rectilinear  System. 

Equations  of  the  Straight  Line. 

25.  General  equation  of  the  first  degree 35 

26.  Common  forms 36 

27.  Derivation  of  the  common  forms  from  the  general  form 38 

28.  Illustrations 39 

29.  Discussion  of  the  common  forms -40 

30.  Construction  of  a  straight  line  from  its  equation.     Examples ...  43 

31.  Equation  of  a  straight  lino  through  a  given  point 45 

32.  Equation  of  a  straight  line  through  two  given  points.    Examples  45 

Plane  Angles. 

33.  Angle  betiveen  two  straight  lines 47 

34.  Equation  of  a  line  making  a  given  angle  witli  a  given  line 48 

35.  Conditions  of  parallelism  and  perpendicularity.     Examples....  49 

Intersections. 

36.  Intersection  of  loci.     Examples 51 

37.  Lines  through  the  intersections  of  loci , .  53 


CONTENTS.  Vll 
Distances  between  Points  and  Lines,  and  Angle-Bisectors. 

ART.  PAGE. 

38.  Distance  of  a  point  from  a  line 55 

39.  Another  method.     Examples 56 

40.  Equation  of  the  angle-bisector.     Examples   58 

Section  V.  —  The  Polar  System. 

41.  Derivation  of  polar  from  rectangular  equations 60 

42.  Polar  equation  of  a  straight  line.     Normal  form.     Examples...  60 

Section  VI.  —  Applications. 

43.  Recapitulation 63 

44.  Properties  of  rectilinear  figures 64 


CHAPTER  III.  —  EQUATION  OF  THE  SECOND  DEGREE,  THE 

CONIC  SECTIONS. 

Section  VIT.  —  Common  Equations  of  the  Conic  Sections. 

45.  The  Conic   Sections 71 

The  Circle. 

46.  Definitions 72 

47.  General  equation  of  the  circle 72 

48.  The  equation  of  every  circle  some  form  of 

y^'J^x:^-\-Dy  +  Ex  +  F=0 73 

49.  Every  equation  of  the  form  ^^  4.  ^2  ^  £)j  ^_  £j.  _^  j^_  q  ^Ij^,  gqug. 

tion  of  a  circle 74 

50.  To  determine  the  radius  and  centre 75 

51.  Concentric  circles.     Examples 75 

52.  Polar  equation  of  the  circle 77 

The  Ellipse. 

53.  Definitions 78 

54.  Central  equation 78 

55.  Definitions 80 

66.   Common  form  of  central  equation 80 

57.  Length  of   focal  radii 81 

58.  Polar  equation 81 

59.  The  ratio   82 


Viii  CONTENTS. 

ART.  PAGE. 

GO.   Geometrical  construction  of  the  ellipse 83 

01.  The  circle  a  particular  case 85 

02.  Varieties  of  the  ellipse.     Examples 85 

The  Hyperbola. 

63.  Definitions 87 

64.  Central  equation 87 

65.  Definitions 89 

66.  Common  form  of  central  equation 80 

67.  Length  of  focal  radii 90 

68.  Polar  equation 91 

69.  The  ratio  9^ 

70.  Geometrical  construction  of  the  hyperbola 93 

71.  The  equilateral  and  conjugate  hyperbolas 95 

72.  Varieties  of  the  hyperbola.     Examples 95 


73. 


The  Parabola. 

Definitions 98 

74.  Common  equation 98 

75.  Polar  equation 99 

76.  Geometrical  construction  of  the  parabola.     Examples 100 

Section  VIII.  —  General  Equation  of  the  Conic  Sections. 

77.  Definitions 102 

78.  General  equation  of  the  conies.     Examples 102 

79.  Every  equation  of  the  second  degree  the  equation  of  a  conic. . . .  103 

80.  Determination  of  species.     Examples 105 

81.  The  equation  ^^2+ Cx2  +  Dj/  + £'.r+jP=0  represents  all  species.  106 

82.  Definitions 108 

83.  Centres 108 

84.  The  equation  Ay^+  Cx^+F-0  represents  all  ellipses  and  hyper- 

bolas    108 

85.  Varieties  of  the  parabola 109 

86.  Definitions HI 

87.  Locus  of  middle  points  of  parallel  chords Ill 

88.  Tangents    at   vertices    of    a    diameter   parallel    to    tlie    cliords 

bisected  by  that  diameter 114 

89.  Definitions 114 

90.  Conjugate  diameters  of  ellipse 114 

01.   Every  straight  line  through  the  centre  of  an  hyperbola  meets  the 

hyperbola  or  its  conjugate   115 


CONTENTS.  IX 

ART.  PAGE. 

92.  Definitions 110 

93.  Conjugate  diameters  of  the  hyperbola 110 

Construction  of  Conies  from  their  Equations. 

94.  By  comparison  with  the  general  equation.     E.xamples 117 

95.  By  transformation  of  a.xes.     E.xamples 119 

96.  By  conjugate  diameters.     Examples 122 

97.  Construction  where  i>  =  0.     PLxamples 125 

General  Theorems. 

98.  Conic  section  through  five  points.     Examples 126 

99.  Intersection  of  conies.     Examples 128 

100.  Definitions 131 

101.  Conies,  similar  and  similarly  placed 131 

102.  Condition  for  two  straight  lines 132 

Section  IX.  —  Tangents  and  Normals. 

103.  Definitions 134 

104.  Equations  of  tangent  and  secant.     Examples 134 

105.  Problems  and  E.xamples 137 

106.  Chord  of  contact 141 

107.  Equation  of  the  normal.     Examples 142 

108.  Definitions 143 

109.  Subtangent  and  subnormal.     Geometrical  constructions 143 

Section  X. — -Oblique  Axes. 
Conjugate  Diameters. 

110.  Equations  of  ellipse  and  hyperbola 150 

111.  Ordinates  to  conjugate  diameters  of  ellipse 151 

112.  Same  for  hyperbola 152 

113.  Parameter  a  tliird  proportional  to  the  axes 152 

1 14.  Circumscribed  circle 152 

115.  Sum  of  the  squares  of  conjugate  diameters  constant 152 

116.  Difference  of  the  squares  of  conjugate  diameters  constant 153 

117.  Eectangle  of  the  focal  radii 154 

118.  Circumscribed  parallelogram 154 

119.  Equal  conjugate  diameters 155 

Supplemental  Chords, 

120.  Definitions 156 

121.  Property  of  supplemental  chords.     Geometrical  constructions. .  156 


CONTENTS. 


Parabola  referred  to  Oblique  Axes. 

ART.  PAGE. 

122.  Equation  of  the  parabola 157 

123.  Property  of  ordinates 159 

Asymptotes, 

124.  Equation  of  the  hyperbola 159 

125.  Property  of  the  secant.     Geometrical  construction 161 

126.  Property  of  the  tangent.     Geometrical  construction 161 

127.  Tangents  meet  on  the  asymptotes 162 


CHAPTER   IV.  — LOCI. 

128.  Classification  of  loci 164 

Section  XI,  —  Loci  of  the  First  and  Second  Order, 

129.  Examples 166 

Section  XTI.  —  Higher  Plane  Loci. 

130.  1.  The  cardioid.     Trisection  of  the  angle 176 

2.  The  conchoid.     Trisection  of  the  angle   177 

3.  Tlie  cissoid.     Duplication  of  the  cube 179 

4.  The  lemniscate 181 

5.  The  witch 182 

6-8.  Examples 183 

Section  XITL  —  Transcendental  Curves. 

131.  1.  The  logarithmic  curve 185 

2.  The  cycloid   185 

3-10,  The  circular  functions  187 

11,  Spiral  of  Archimedes 189 

12,  Reciprocal  spiral ; 190 

13.  The  lituus 191 

14.  Logarithmic  spiral 192 


CONTENTS.  XI 


Part  IL  —  Solid  Analytic  Geometry. 


CHAPTER  v.— THE   POINT,  STRAIGHT   LINE,   AND    PLANE. 

Section  XIY.  —  Introductory   Theorems. 

ART.  PAGE. 

132.  Definitions.     Projections  of  lines 195 

133.  Length  of  the  projection  of  a  line  on  a  line  and  plane 196 

134.  Projection  of  a  broken  line 197 

Section  XV.  —  The  Point. 

1.35.   Position  of  a  point 198 

130.   Equations  of  a  point.     Examples 199 

137.  Distance  between  two  points    200 

138.  Polar  coordinates  of  a  point 201 

139.  Relations  between  polar  and  rectangular  coordinates 201 

140.  Direction-angles  and  -cosines 202 

141.  Relation  between  direction-cosines.     Examples 202 

142.  Angle  between  straight  lines 203 

Section  XVI.  —  The  Plane. 

143.  General  equation  of  a  surface 205 

144.  Equation  of  a  plane 206 

145.  Intercept  form.     Examples 207 

146.  Normal  form.     Examples 208 

147.  Equation  of  a  plane  through  three  points.     Examples 209 

148.  Angle  between  planes.     Examples 210 

149.  Traces  of  a  plane 213 

Section  XVII.  —  The  Straight  Line. 

150.  Equations  of  a  straight  line /. 214 

151.  Symmetrical  forms ". 210 

152.  Reduction  to  symmetrical  form.     Examples 216 

153.  Equations  of  a  line  through  two  points 218 

154.  Angle  between  straight  lines.    Examples  and  problems 219 


Xll  CONTENTS. 


CHAPTER  VI.  — SURFACES  OP  REVOLUTION,   CONIC 
SECTIONS,  AND   THE  HELIX. 

Section  XVIII.  —  Surfaces  of  Revolution. 

ART.  PAGE. 

155.  Definitions 222 

156.  General  equation  of  a  surface  of  revolution 222 

157.  The  sphere 223 

158.  The  prolate  spheroid 223 

159.  The  oblate  spheroid ...    224 

160.  Tlie  paraboloid 224 

161.  Tlie  liyperboloid  of  two  nappes 224 

162.  The  hyperboloid  of  one  nappe 225 

163.  The  cylinder 225 

164.  The  cone 226 

Section  XIX.  —  The  Conic  Sections, 

165.  General  equation  of  the  plane  section  of  a  cone,  and  its  discus- 

sion    227 

Section  XX.  —  The  Helix. 

166.  Definitions 229 

167.  Equations  of  the  helix 229 


PART   I. 
PLAICE   Al^ALYTIC   GEOMETEY. 


CHAPTER   T. 
COORDINATE    SYSTEMS. 


3j«<0<^ 


SECTION  I. —THE   POINT. 


THE    RECTILINEAR    SYSTEM. 

1.    Position  of  a  point  in  a  plane.      The  usual  method  of 

locating  a  point  on  the  earth's  surface  is  by  its  latitude  and 

longitude,   reckoned  respectively  from  the   equator  and  some 

assumed  meridian.     A  similar  method  serves  to  fix  the  position 

of  a   poiut  in   a  plane.      Thus:    if  X'X,    Y'Y,   be    any   two 

assumed  straight  lines  intersecting  at  0,  the  position  of  a  point 

P'  in  their  plane,  with  reference  to 

these  lines,  will  be  known  when  the  pn  /  pi 

distancesmP^iPf,  are  known,  these  /  J"  J 

distances  being  measured  parallel  to         ,w  /  /o        ml 

°  *-  A—/ 4 -^ — -x, 

Y'  Y  and  X'X.     If,  however,  only  /  /  / 

the  numerical  lengths  of  onPK  7iP\    ^,^/  /  / 

are  known,   the  point   may   occupy  /y' 

any  one  of  four  positions,  P\  P",  Kig  i. 

P™,  P"'.  This  ambiguity  will  dis- 
appear if  we  distinguish  distances  laid  off  above  and  below  X'X, 
parallel  to  I"'  Y,  as  positive  and  negative,  respectively  ;  and,  in 
like  manner,  as  positive  and  negative,  respectively,  those  laid 
off  to  the  right  and  left  of  Y'Y,  parallel  to  X'X.  Hence,  the 
position  of  0719/  point  in  the  plane  of  X'X  and  Y'Yicill  be  com- 
pletely determined  tvith  reference  to  these  lines  when  its  distances 
from  them,  measured  parallel  to  them,  are  given  in  magnitude 
and  sign. 


Z  ANALYTIC    GEOMETEY. 

2.  Defs.  The  fixed  lines  X'X,  Y'Y,  are  called  the  axes  of 
reference;  their  intersection.  0,  the  origin;  the  distances  mF\ 
nF\  the  coordinates  of  the  point  P' ;  and  to  distinguish  these  co- 
ordinates, nP^  is  called  the  abscissa,  and  mP'  the  ordinate  of  P\ 

3.  Construction  of  a  point.  Since  the  coordinates  of  a  point, 
when  given,  fix  its  position  with  reference  to  the  axes,  and 
since  (Fig.  1)  Oin  =  nP\  On.  =  mP\  to  determine  the  position  of 
a  point  whose  coordinates  are  given  we  have  simply  to  lav  off  the 
given  abscissa  from  0  along  X'X  in  the  direction  indicated  by 
its  sign,  and  at  its  extremity  on  a  parallel  to  F'F,  the  given 
ordinate,  above  or  below  X'X  according  as  it  is  positive  or 
negative.  This  determination  of  the  position  of  a  point  is 
called  the  construction  of  the  point. 

4.  The  axes  of  reference  are  always  lettered  as*  in  Fig.  1, 
and  hence  are  often  designated  as  the  axes  of  X  and  5",  the 
former  being  usually  taken  horizontal.  For  brevity  they  will 
frequently  be  spoken  of  as  X  and  Y  simpl}-.  The  abscissa  of 
a  point,  being  always  a  distance  parallel  to  the  axis  of  X,  is 
always  represented  b}^  the  letter  x ;  and  for  a  like  reason  the 
ordinate  is  always  represented  by  the  letter  y  ;  hereafter,  there- 
fore, these  letters  will  always  represent  distances  parallel  to 
the  axes  of  reference.  As  indicating  the  directions  in  which 
abscissas  and  ordinates  are  laid  off,  the  axes  are  also  distin- 
guished as  the  axis  of  abscissas  {X'X).  and  the  axis  of  ordi- 
nates {Y'Y).  The  angles  XOY,  YOX',  X'OY',  Y'OX,  are 
known  as  the  first,  second,  third,  and  fourth,  angles,  respectively. 

It  is  evident  that  so  long  as  the  angle  XOY  is  not  zero,  it 

may  have  any  value  whatever.     When 

/p'  a  right   angle,  the  system  of  reference 

i  is   called   a  rectangular  system;  other- 

Q  j,„        ^      wise,  an  oblique  system.     As  nothing  in 

general,  is  gained  by  assuming  oblique 

axes,  the  axes  will  hereafter  be  supposed 

Y'  rectangular,  unless  mention  is  made  to  the 

Fig.  2.  contrary  ;  the  abscissa  and  ordinate  of  a 


Y 

n 


X^ 


THE   POINT.  6 

point  will  thus  be  (Fig.  2)  the  perpendicular  distances  of  the 
point  from  the  axes. 

5.  Equations  of  a  point.  It  is  now  evident  that  we  ma}' 
designate  the  position  of  a  point  by  giving  its  coordinates  in  the 
form  of  equations.  Thus,  .r  =  2,  ?/  =  3,  designate  a  point  in  the 
first  angle,  distant  3  units  from  the  axis  of  X,  and  2  units  from 
the  axis  of  Y.  These  equations  are  called  the  equations  of  the 
point.  But  it  is  more  usual  to  adopt  the  notation  (2,  3)  to 
designate  the  point,  the  abscissa  being  always  loritten  first. 

Examples.  1.  What  are  the  signs  of  the  coordinates  of  all 
points  in  the  first  angle?  In  the  second?  In  the  third?  In 
the  fourth  ?  Where  are  all  points  situated  whose  ordinate^are 
zero?  Whose  ordinates  are  equal  and  have  the  same  sign? 
Whose  abscissas  are  zero?  What  are  the  coordinates  of  the 
origin  ? 

2.  Construct  the  following  points  :  (2,4);  (3,-2);  (  —  6, 
-2)  ;    (-4,  3)  ;   (2,  0)  ;   (0,  -2)  ;   (-2,  0)  ;   (0,  2)  ;   (0,  0). 

3.  Construct  the  triangle  whose  vertices  are  (4,  5),  (4,  —5), 
(  —  4,  5).  What  kind  of  a  triangle  is  it,  and  what  are  the 
directions  of  its  sides  ? 

4.  The  side  of  a  square  is  a,  and  its  centre  is  taken  as  the 
origin,  the  axes  being  parallel  to  its  sides.  What  are  the  coor- 
dinates of  the  vertices?  What,  when  the  axes  coincide  with  the 
diagonals,  the  origin  being  still  at  the  centre? 

5.  An  isosceles  triangle,  whose  base  is  h  and  altitude  a,  has 
its  base  coincident  with  X.  What  are  the  coordinates  of  its 
vertices  when  the  origin  is  (1)  at  the  centre  of  the  l)ase?  (2)  at 
the  left  hand  extremity  of  the  base  ? 

6.  Construct  and  name  the  figures  whose  vertices  are 

(1).  (a,  a),  («,-«),  (-ff,  -«),  {-a,  a). 

(2).  {a,b),  (a,  -b),  (-a,  -6),  (-«,&).'■ 

(3).  (a,&),  (o,  -&),  (-a,  -6),  {-a,c). 

(4).  (o,  &),  {c,d),  (-e,  d),  (-/,&). 


ANALYTIC   GEOMETIIY. 


6.  To  find  the  coordinates  of  a  point  ichich  divides  the  straight 
line  joining  two  given  j^oints  in  a  given  ratio. 

Let  P',  P",  be  the  given  points,  x',  y\  and  a*",  3/",  their 
coordinates,  and  P  a  third  point  {x,  y),  dividing  P'P"  so  that 
P'P  :  PP"  ::  m:n.     Then,  from  similar  triangles, 

Y  P" 

(1) 


P'P      BP 


PQ      QP" 


n 


Bnt  P'E  =x-x',PQ  =  x"-  x,  RP=y-  y', 
QP"=  y"—  y.     Substituting  these  values, 


Fig.  3. 


X  —  x' _y  —  y' _  m 
x"—x     y"  —  y      n 


Solving  these  equations  for  x  and  y,  we  obtain 


x  = 


mx"  -\-nx' 
m  +  n 


my"  -\-ny' 
m  -j-  n 


(2) 


For  the  middle  point  of  P'P",  m  =  n.     Hence  the  coordinates 
of  the  middle  point  of  a  line  joining  two  given  points  are 


X  — 


x"-\-x' 
2      ' 


y  = 


y"+y' 
2 


(3) 


If  the  line  is  cut  externally,  we  have  from 
(1)  and  Fig.  4, 


X  —  x'  _  y  —  y' 
X  —  x"      y  —  y" 


m 

n 


whence 


0 


Kii,'.  4. 


X 


mx"—nx' 
m  —  ?i 


y 


my"  —  ny' 
m  —  n 


(4) 


Oblique  Axes.    Tlie  above  formulas  hold  good  for  oblique  .ixcs,  since  the  triangles 
remain  similar  whatever  the  angle  XO  Y. 

Examples.     1     Find  tlie  coordinates  of  the  middle  point  of 
the  line  joining  (5,  3)  and  (3,  9);  also  (5,  8)  and  (  —  5,  —8). 

Ans.    (4,  G);   (0,  0). 


THE   POINT.  O 

2.  Find  the  coordinates  of  the  middle  points  of  the  sides  of 
the  triangle  whose  vertices  are  (G,  2),  (8,  4),  (10,  12). 

Ans.   (7,  3);   (9,  8):   (8,  7).' 

^'3.  The  line  joining  (  —  2,  —3)  and  (  —  4,  5)  is  trisected. 
Find  the  coordinates  of  the  point  of  trisection  nearest  (  —  2,  —3). 

154.  The  line  whose  extremities  are  (2,4)  and  (G,  —8)  is 
divided  in  the  ratio  3:2.  Find  the  two  points  of  division  ful- 
filling the  condition.  >         /22         1G\      /1 8         4 

Qo.  Find  the  two  points  on  the  line  joining  (2,  4)  and  (6,  3) 
twice  as  far  from  (2,  4)  as  from  (G,  3). 

Ans.    (10,  2);   f— ,    - 

G.  The  vertices  of  a  triangle  are  (  —  4,  —3),  (6,  1),  (4,  11). 
Find  the  coordinates  of  the  points  of  trisection,  farthest  from 
the  vertices,  of  the  lines  joining  the  vertices  and  the  middle 
points  of  the  opposite  sides.  A71S.    (2,  3). 

7.    To  find  the  distance  between  two  given  j^oints. 

Let  P',  P",  be  the  given  points,  x',  y',  and  x'\  y",  being  their 
coordinates.  Then  FT"  =  VP'B'  +  BP"- ;  or,  representing 
P'P"  bv  rf, 


d  =  V{x"-x'r  +  {ij"-!j'y.      (1) 

If  one  of  the  points,  as  P",  is  at  the  ori- 
gin, its  coordinates  will  be  zero.  Hence  the 
distance  of  any  point  P'  from  the  origin  is 


d  =  ^x"  +  y".  (2) 


Fig.  5. 


|5  Oblique  Axes.    In  this  case  the  triangle  P' RP"  will  not  be  a  right  triangle. 
Let  ^  =  inclination  of  the  axes.    Then 

P'P"=  y/PIt^  +  RP'"^-2P'P  .  HP"  COS  P' HP", 

or,  since  P'PP"  =  ISO^  -  3, 


d=  V(a;"-a;')^+(2/"-2/')-  +  2(a;"-a;')  {y"-y')coBp,  (3) 

which,  when  ^  =  90^,  reduces  to  (1),  since  cos  90°  =0. 


6  ANAXYTIC    GEOIVLETRY. 

Examples.       1.    Find    tlie    distance    between     (2,  4)    and 

(5,8). 

As  the  quantities  under  the  radical  sign  are  squares,  it  is  immaterial 
whether  we  substitute  (2,4)  for  {x',  i/')  and  (5,  8)  for  {■'-",>/"),  or  vice 
versa.     Thus 

(/  =  V'(2-5)--2  +  (4-8)^  =  V(5-2)--2+(8-4)''2  =  5. 

2.  Find  the  distances  between  the  following  points  :  (—2,  —4) 

and  (-.'),  -8);   (7,  -1)  and  (-G.  1);   (7,  2)  and  (  —  7,  -2).  ^r^TT" 

Alls.  .') ;  Vl73  ;  2  V53. 

3.  Find  the  distance  of  (G,  —8)  from  the  origin.       Ans.   10. 

4.  Find  the  lengths  of  the  sides  of  the  triangle  whose  vertices 
are  (4,  8),  (1,  4),  (-4.  -8).  j^^s.  5;  13;  8V5. 


5.  Find  the  lengths  of  the  sides  of  the  triauole  whose  vertices 
are  (4,  5),  (4,  -o),  (-4,  5).  Ajis.   10;  2  V4l ;  8. 


THE    POLAR    SYSTEM. 

8.  Position  of  a  point  in  a  plane.  The  position  of  a  point 
on  the  earth's  surface  is  often  designated  by  its  distance  and 
direction  from  some  other  point ;  as  when  A  is  said  to  be  25 
miles  northeast  of  B.  In  a  similar  way  the  position  of  a  point 
in  a  plane  may  be  designated.     Thus  :  if  OA  be  an}'  assumed 

straight  line  through  a  fixed  point  0, 

yP'  the  position  of  a  point  P'  in  the  plane 

/^  AOP',    with   reference   to  0.  will   be 

/        '  ^      knoAvn  when  the  angle  AOP'  and  the 

/  distance   OP'  are  known.      The  fixed 

P" 

Fig..;.  line  OA  is  called  the  Polar  Axis;  the 

fixed    point,   O,  the   Pole;    the    angle 

AOJ"  and  distance  OP',  the  Polar  Coordinates,    OP'  being  the 

radius  vector  and  AOP'  the  vectorial  angle.     The  ratlins  vector 

will   always  be  represented  by  the  letter  ?•,   and  the  vectorial 

angle  by  the  letter  &. 


u 


THE    POINT.  7 

9.  Signs  of  the  polar  coordinates.  If  the  vectorial  angle  be 
always  l:iid  off  above  OA  (Fig.  6)  to  the  left,  as  iu  trigonom- 
etry, the  position  of  every  point  in  the  plane  may  be  desig- 
nated without  ambiguity,  and  were  this  the  only  consideration 
there  would  be  no  necessity  for  any  convention  as  to  signs. 
But  as  r  and  6  often  occur  in  the  course  of  analytic  investiga- 
tions with  negative  as  well  as  positive  signs,  it  is  necessary  to 
adopt  some  convention  for  the  interpretation  of  the  negative 
sign.  For  this  purpose  the  vectorial  angle  is  regarded  positive 
when  laid  off  above  OA  to  the  left,  and  negative  when  laid  off 
below  OA  to  the  right ;  while  the  radius  vector  is  considered 
positive  when  laid  off  from  O  towards  the  end  of  the  arc  measur- 
ing the  vectorial  angle,  and  negative  when  laid  oft"  in  the  oppo- 
site direction.  Thus  (Fig.  6),  for  the  angle  AOP',  OF'  is  the 
positive,  and  OP"  the  negative,  direction  of  r. 

10.  Construction  of  a  point.  Since  the  coordinates  r  and  0, 
when  given,  fix  the  position  of  a  point,  to  determine  its  position 
we  have  only  to  hi}"  oft"  the  given  value  of  0  above  or  below 
OA  (Fig.  6)  according  as  0  is  positive  or  negative,  and  on  the 
line  through  0  and  the  end  of  the  measuring  arc  the  given 
value  of  r,  towards  or  away  from  the  end  of  the  measuring 
arc  as  the  sign  of  r  is  positive  or  negative.  This  determina- 
tion of  the  position  of  a  point  is  called  the  construction  of  the 
2Mint. 

11.  Equations  of  a  point.  It  is  now  evident  that  we  may 
designate  the  position  of  a  point  bj'  giving  its  coordinates  in  the 
form  of  equations.  Thus  (Fig.  G),  ?-  =  4,  ^  =  60°,  locate  a 
point  P'  distant  4  units  from  0  on  a  line  inclined  at  -|-  60°  to 
OA;  while  r  =  — 4,  ^  =  00°,  locate  a  point  P"  on  the  same 
line,  but  on  the  opposite  side  of  0.  These  equations  are 
called  the  jwlar  equations  of  a  jyoint;  but  it  is  more  usual 
to  adopt  the  notation  (r,  ^),  writing  the  radius  vector  first. 
Thus,  the  above  points  would  be  (4,  GO")  and  (—4,  G0°), 
respectively. 


8  ANALYTIC   GEOMETRY. 

12.  This  system  of  reference  is  called  the  Polar  System,  aud 
that  previously  described,  whether  the  axes  be  oblique  or  rec- 
tangular, the  Rectilinear  System.  It  will  be  observed  that  in 
each  s^'stem  tico  things  serve  as  the  bases  of  reference  ;  in  the 
rectilinear,  the  axes  of  X  and  Y\  in  the  polar,  the  pole  and 
polar  axis.  Also  that  in  each  system  tivo  quantities  are  suffi- 
cient to  refer  the  point ;  in  the  rectilinear  system,  the  abscissa 
and  ordinate ;  in  the  polar,  the  radius  vector  and  vectorial 
angle.  Again,  that  while  in  the  rectilinear  system  a  given  point 
can  have  but  one  set  of  coordinates,  in  the  polar  S3^stem  it  may 
have  an  infinite  number  of  sets.  Thus  (Fig.  6),  P'  may  be 
designated  as  follows:  (4,60°),  (-4,  -120°),  (-4,  240°), 
(4,  -300°),  (4,  420°),  etc.  This  fact,  however,  gives  rise 
to  no  ambiguity  in  the  position  of  P',  for  no  one  set  of  polar 
coordinates  can  locate  more  than  one  point. 

The  rectilinear  aud  polar  systems  of  reference,  together  with 
a  third  called  the  trilinear,  are  those  in  most  common  use. 
The  two  former  only  will  be  employed  in  this  treatise. 

Examples.    1.    Construct   the    following    points:    (o,   90°); 

(5,270°);   (-3,  120°);    (-G,  -180°). 

2.  What  are  the  coordinates  of  the  pole?  AYhat  are  all 
possible  values  of  0  for  points  on  the  polar  axis? 

3.  Construct  the  points  (0,  45°)  ;  fo,  -\  ;  (4,0°)  ;  (-4,0°). 

4.  Give  three  sets  of  polar  coordinates  locating  (10,  90°). 

5.  Construct  (s.^^Y,  f-S,  ^)  ;  fs,  ^)  ^  (§'  "  'f 

6.  The  side  of  a  square  is  5V2,  its  centre  at  the  pole,  and 
sides  parallel  and  perpendicular  to  the  polar  axis.  AVhat  are 
the  coordinates  of  its  vertices? 

P"  13.    To  find    the    distance    between    two 

I  ^^p'  (jiven    points.     Let   P',  P",    be   the    given 

points,  r\  0\  and  r",  0",  their  coordinates, 
^    and  d  the  required  distance.     Then,  from 


/ 


Fi^'.  7.  the  triangle  P'OP", 


THE   POINT. 

P'P"=  VOP'--f-  OP"'-  2  OF.  OP"  COS  P'OP", 


or  f?=  Vr'2  +  r"2-2/r''"cos  (6^"— 6').  (1) 

If  oue  of  the  points,  as  P'\  is  at  the  origin,  d  —  OP—  r' . 

Examples.    1.    Fiud  the  distance  between  the  points  (3,^ 
and  (  4.  -'^ 


o 

Since  cos  6  =  cos  (—  6) ,  it  is  immaterial  which  of  the  two  given  points 
is  designated  as  (c',  6') .     Thus 

d=  V9+ 16-24  cos  00°=  VlO^  9-24  cos  (-60")  =  VlS. 
Observe,  also,  that  if  6''—  6'  >  90°,  the  cosine  will  be  negative  and  the  last 
term  positive,  as  it  should  be,  for  then  the  triangle  will  be  obtuse-angled 
at  0  (Fig.  7). 

2.    Find  the  distances  between  the  following  points  :   (3,  60°) 
and  (4,  150°)  ;    (5,  0°)  and  (5,-180°)  ;  (^2,-)  and  (1,  0°)  ; 

(10,  30°)  and  (-10,  -  150°)  ;   (G,  G0°)  and  (0,  0°) 

Arts.  5  ;  10  ;  1  ;  0  ;  6. 


10 


ANALYTIC   GEOxMETKV, 


SECTION    II.— THE   LINE. 


THE    RECTILINEAR    SYSTEM. 


14.  Loci  and  their  equations.  Every  line,  straight  or  curved, 
may  he  regarded  as  generated  by  the  motioi  of  a  j'oint.  The 
kind  of  line  generated  will  depend  upon  the  law  which  governs 
the  motion  of  the  generating  point.  Thus,  a  circle  may  be 
traced  by  a  moving  point,  the  law  which  governs  its  motion 
being  that  it  shall  always  remain  at  a  given  distance  (the  radius) 

from  a  fixed  point  (the  centre).  If 
the  origin  be  taken  at  the  centre  of 
the  circle,  P  being  any  point  of  the 
circle,  x,  y,  the  coordinates  of  P,  and 
0P=  B,  the  radius,  then 

0P-=  Om-  +  mI'^, 
or  X-  +  y-  =  7?-, 

is  true  for  every  position  of  P  while 

generating  the  circle.     This  equation 

is  the  algebraic  expression  of  the  law 

which  governs  P's  motion,  and  is  called  the  equation  of  the  circle ; 

and,  in  genei-al,  the  equation  of  a.  line  is  the  algebraic  expressitm 

of  the  law  ivhich  governs  the  motion  of  its  generating  point. 

Again,  the  relation  x-  +  y'  =  R'  is  true  for  no  point  within  or 
without  the  circle,  ])ut  is  true  for  every  point  on  the  circle  ;  it 
thus  expresses  the  relation  between  the  coordinates  of  all  points 
of  the  circle,  and  of  no  other  jioints  ;  hence,  in  general,  the 
equation  of  a  line  is  the  algebraic  exjjression  of  the  relation  ivhich 
exists  between  the  coordinates  of  any  and  every  point  of  the  line. 
Evidently  if  a  point  moves  at  random,  without  any  governing 
law,  the  line  it  traces  can  have  no  equation  ;  for  the  latter  is 


THE    LINE.  .  11 

the  algebraic  expression  of  a  law,  and  when  the  point  moves  at 
random,  none  such  exists. 

The  above  equation,  x-  +  y-  =  B^,  being  the  equation  of  a 
circle  whose  radius  is  li,  the  circle  is  said  to  be  the  locus  of  the 
equation;  i.e..  translating  the  word  locus  literally,  it  is  the 
2)lare  in  which  the  point,  moving  under  the  law  expressed  by 
the  equation,  is  always  found  ;  and,  in  general,  the  locus  of  an 
equation  is  the  2)ath  of  <(  poiid  so  moving  that  its  coordinates 
always  fulfil  the  relation  e.vpressed  by  the  equation. 

It  follows  that  if  a  point  lies  on  a  locus,  the  coordinates  of 
the  point  must  satisfy  the  equation  of  the  locus.  Thus,  if  the 
radius  of  the  above  circle  be  ."j,  x*- +  7- =  25,  and  the  points 
(3,  4) ,  (0,  —  5) ,  ( —  4,  —  3) ,  are  all  points  on  the  circle  because 
their  coordinates  satisfy  the  equation  ;  but  (2,  4),  (—  4,  4),  are 
not  on  the  circle.  Hence,  to  ascertain  tvhether  a,  given  point  ties 
on  a  given  locus,  substitute  its  coordinates  in  the  equation  of  the 
locus  and  see  whether  they  satit<f/  it. 

15.  Distinctions  between  Analytic  Geometry,  Geometry,  and 
Algebra.  The  object  of  Analytic  Geometry  /.s  tJie  discassion  and 
determination  of  the  prop)erties  of  loci.  Its  method  consists  in 
the  substitution  of  the  equation  of  the  locus  for  the  locus  itself 
in  the  discussion  and  determination  of  its  properties.  Thus, 
X-  -\-y-  —  R^  has  been  seen  to  be  tlie  equation  of  the  circle  of 
Fig.  8.     Putting  it  under  the  form 

f-  =R--  X-  =  (/?  +  X)   (E  -  .V) , 

we  observe  that 

/  =  Pm-,  It  +  a-=  xi'm,  R  -  x  =  mA. 

Hence  Pm- =  A'm.mA.,  or  the  square  of  the  half-chord  to  any 
diameter  of  a  circle  is  a  mean  proportional  between  the  seg- 
ments into  which  it  divides  that  diameter.  This  well-known 
property  of  the  circle  might  be  established  geometriccdly ,  from 
a  figure  ;  it  is  here  established  analyticcdly,  from  the  equation 
of  the  circle  ;  and  the  object  of  Analytic  Geometry  is  thus  to 
determine  the  properties  of  lines,  by  discussing  their  equations, 


12  ANALYTIC    GEOMETRY. 

instead  of  b}'  reasoning  upon  the  lines  tliemselves  as  in  Eucli- 
dean Geometry. 

Having  thus  noted  the  distinction  between  Analytic  Geometry 
and  Geometry,  let  us  note  iu  what  way  it  differs  from  Algebra. 
Since  the  coordinates  of  every  point  of  the  circle  must  satisfy 
its  equation  x-  -{-?/-  =  Ii\  x  and  y  in  this  equation  may  have  an 
infinite  number  of  sets  of  values,  corresponding  to  the  infinite 
number  of  positions  occupied  b}'  the  generating  point  in  tracing 
the  circle.  Hence  while  i^  is  a  constcmt  quantity,  x  and  y  are 
variable  quantities.  They  differ  thus  from  all  the  quantities 
of  common  Algebra,  which,  whether  known  or  uukuown,  are 
always  constants.  Observe,  also,  that  while  x  and  y  thus  admit 
of  an  infinite  number  of  values,  they  do  not  admit  of  any  val- 
ues, but  only  of  those  which  satisfy  the  relation  x^  -\-  y-  =  JR.^. 
Again,  in  Algebra,  if  only  x-  -\-  y-  =  R^  were  given,  x  and  y  being 
unknown  but  constant  quantities  wliose  values  were  required, 
the  solution  would  be  impossible  ;  for  this  equation  would  be 
satisfied  by  an  infinite  number  of  sets  of  values  of  x  and  y.  and 
without  a  second  independent  equation  we  could  not  determine 
the  particular  values  required.  Furthermore,  if  the  conditions 
of  the  problem  were  not  such  as  to  furnish  a  second  equation, 
the  problem  would  remain  an  indeterminate  one.  It  is  in  virtue 
of  this  very  indetermination  that  we  are  enabled  to  represent 
loci  by  equations,  and,  as  thus  distinguished  from  Algebra, 
Analytic  Geometry  is  sometimes  called  the  Indeterminate 
Analysis. 

16.  Quantities  of  Analytic  Geometry. 
If  tlic  centre  of  the  circle  were  at  some 
point  C,  whose  coordinates  are  m  and  n ,  in- 
stead of  at  the  origin,  then,  from  tlie  right- 
angled  triangle  PCE,  CR"+liP-=CP", 


or 


^x-my  +  {y-ny-  =  E\ 


This  relation  between  x  and  y,  being 
true  for  all  positions  of  P  on  the  circle, 
is  the  equation  of  the  circle  in  its  new 


THE    LINE.  13 

position  witli  reference  to  the  axes.  Now  if,  in  this  equation 
of  the  circle,  we  change  7?,  we  change  the  magnitude  of  the 
circle  ;  and  if  we  change  m,  or  n,  or  both,  we  change  the  position 
of  the  circle.  Hence  ~^/<e  constants  in  the  equation  of  a  locus 
determine  the  magnitude  and  position  of  the  locus.  The  quan- 
tities of  Analytic  Geometry  are  thus  : 

First.  Variable  quantities^  as  x  and  y  of  the  preceding  equa- 
tion, which,  being  the  coordinates  of  a  moving  point,  vary  con- 
tinuously within  the  limits  assigned  by  the  equation  expressing 
their  mutnal  relation.  Thus  a;  varies  continuous!}'  between  the 
limits  X  =  OJ/,  and  x  =  ON.,  and  y  between  the  limits  y  =  QS, 
and  y  =  QT.  Since,  when  y  changes,  x  also  changes,  and  vice 
versa.,  x  and  y  are  said  to  be  functions  of  each  other. 

Second.  Arbitrary  constajits,  as  m^  ?;,  B,  of  the  above  equa- 
tion, to  which  values  may  be  assigned  at  pleasure,  thus  locating 
any  circle  in  any  position.  They  do  not,  however,  change  lohen 
X  and  y  change.,  that  is,  tlie}'^  are  not  functions  of  x  and  y,  and 
are  thus  constants,  though  arbitrary  constants. 

Third.  Absolute  C07istants,  such  as  m,  n.  and  R,  would 
become  in  the  above  equation,  if  we  placed  the  centre  of  the 
circle  at  (7,  G)  and  assumed  5  for  its  radius;  which  cannot 
change  under  any  circumstances. 

17.  Construction  of  loci.  Tt  is  now  evident  that  two  general 
classes  of  problems  will  arise. 

First.  Given  the  laio  governing  the  motion  of  the  generating 
point  (usually  given  in  the  form  of  some  property  of  the  locus) , 
to  find  the  equation  of  the  locus. 

Second.  Given  the  equation  of  the  locus,  to  determine  the 
locus;  i.e.,  its  position,  form,  and  properties. 

These  two  fundamental  problems  form  the  subject  matter  of 
Analytic  Geometry  and  will  be  fully  illustrated  in  the  sequel. 
Their  solution  involves  on  the  part  of  the  student  a  thorough 
comprehension  of  the  relation  between  a  locus  and  its  equation 
as  defined  in  Art.  14,  and  to  illustrate  this  relation  the  following 
examples  of  the  determination  of  loci  from  their  equations  by 
points  are  added. 


14  ANALYTIC    GEOMETRY. 

Examples.  If  any  value  of  either  variable,  assumed  at 
pleasure,  be  substituted  in  the  equation  of  a  locus,  and  the 
value  of  the  other  variable  be  found  from  the  equation,  the 
set  of  values  thus  obtained  evidenth'  satisfies  the  equation ; 
the}'  therefore  determine  a  point  of  its  locus.  Hence,  to  deter- 
mine points  of  the  locus  of  an  equation,  assume  in  succession 
any  number  of  values  for  one  variable,  and  find  from  the  equa- 
tion the  corresponding  values  of  the  other.  Construct  the 
points  thus  obtained  and  draw  a  line  through  them.  This  line 
will  be  the  locus  of  the  equation.  This  process  is  called  the 
construction  of  the  locus.  The  variable  to  which  values  are 
assUjned  is  called  the  independent  variable;  the  other,  whose 
values  are  derived  from  the  equation,  is  called  the  dejyendent 
variable.  It  is  evident  from  the  nature  of  the  process  that 
either  variable  may  be  chosen  as  the  independent  variable,  and 
it  is  usual  to  assign  values  to  x  and  derive  those  of  y.  In  such 
an  equation,  however,  as  x  =  '>/'—2y^-\-  4,  it  is  more  convenient 
to  assign  values  to  y  and  derive  those  of  x;  i.e.,  to  make  the 
variable  which  is  most  involved  the  independent  variable.  The 
illustrations  which  follow  are  limited  to  equations  of  the  first 
and  second  degree. 

!.■  I/  — .r  — 4  =  0.  Solving  the  equation  for  y,  we  have 
?/  =  x-{-  4,  and  taking  x  for  the  independent  variable,  we  obtain 
for 


Vie.  Id. 


X  =  0, 

2/  =  4, 

locating  P', 

x=  1, 

y=5. 

locating  P", 

x=2, 

2/=6, 

locating  F'", 

x=d, 

2/ -7, 

etc. 

x  =  -\. 

3/=3, 

x  =  -2, 

2/  =  -^ 

x  =  -S, 

y  =  u 

.r  =  —  4, 

!/  =  0, 

x  =  —  ;'). 

(■t( 

.'/  =  -  1 

» 

Constructing  these  points,  the  line  MX  drawn  tluough  tlieni  is 
the  locus  of  y  —  x  —  4  =  0. 


THE    LINE. 


15 


It  will  subsequently  Ije  shown  that  the  locus  of  every  equa- 
tion of  the  lirst  degree,  between  x  and  y,  is  a  straight  line. 
This  being  the  case,  it  is  necessary  to  construct  but  two  points 
for  such  equations.  Assuming  this  fact,  the  student  may  con- 
struct the  straight  lines  represented  by  the  next  four  equations, 
constructing  in  each  case  two  points  ;  then  verif}-  the  construc- 
tion by  locating  a  third  point. 

2.  y  -{-  X—  1  =  0.     Solvnig  for  y,  y  —  —  .f  +  1  ;   in  which,  for 
.«  =  0,       //  =  1 ,       locating  P', 
X  =  5,       y  =  —  4,    locating  P". 
x  —  —  i^,  y  —  \^       locating  P'". 

Constructing  P\  (0,  1),  and  P". 
(5,  —4),  P'P"  should  pass  through 
P'",  (-0,  4). 

3.  ?/  — .f  =  0.  4.    ?/  +  .r=0. 

5.  ?,y  —  '2x  —  \  =  0. 

6.  ?/-=4.c  — 8.  Solving  for  y,  we  have,  j/  =  ±V4.r  — 8. 
Assigning  values  to  .r,  for  x  =  0  we  have  y  =  ±  V  — 8,  which  is 
imaginary  ;  moreover  y  will  evidently  be  imaginary  for  all  values 
of  a;<2,  algebraically.  As  the  ordinates  corresponding  to  all 
values  of  x  <  2  are  imaginary,  we  conclude  that  there  are  no 
points  of  the  locus  having  abscissas  less  than  2  ;  and,  in  gen- 
eral, when  either  of  the  eoordinafes  obtained  from  the  equation  is 
imaginary,  v:e  conchide  there  is  no 
corresponding  j^oint  of  the  locus. 
Assuming  values  for  a;>2,  we 
have,  for 

x=  2,  ^  =  0,  locating  P\ 

x  =  3,  y  =  ±'2,        loc.  P"  and  P", 
X  =  4:,  y  =  ±  2  V2,  loc.  P''  and  7^^, 


Fis.  n. 


X 


5,  7/  =  ±2V3, 


etc. 


x=G,  y=±-i, 

etc., 
every  value  of  x  >  2  locating  two  points 


Fig.  12. 


16  ANALYTIC    GEOMETRY. 

The  above  method  of  constructiug  a  locus  b}'  points  is  a  purely 
mechanical  one.  The  greater  the  number  of  points  located,  the 
more  accurate  the  construction  of  the  locus.  A  simple  inspection 
of  the  equation  will,  however,  often  indicate  the  general  form  and 
position  of  the  locus.  Thus,  in  the  above  example,  every  value 
of  X  gives  two  values  of  ?/  numerically  equal  with  opposite 
signs,  and  the  locus  is  therefore  made  up  of  pairs  of  points 
equidistant  from  X'X,  or  the  axis  of  X  is  an  axis  of  symmetry ; 
and,  in  general,  tchenever  the  equation  contains  the  square  only 
of  either  variable,  the  other  axis  is  an  axis  of  symmetry.  Thus 
y-=^x  is  symmetrical  with  reference  to  X\  y-\-of=2  is  sym- 
metrical with  respect  to  Y;  while  x~-{-'ir=2o,  c(r—y-=i, 
9. r^+ 16.^^=144,  are  symmetrical  with  respect  to  both  coordi- 
nate axes. 

Again  :  since  the  ordinate  of  every  point  on  the  axis  of  X  is 
zero,  if  the  locus  has  any  point  on  the  axis  of  X  it  will  be 
found  by  making  y  =  0  ;  and  for  a  like  reason  if  it  has  any 
point  on  the  axis  of  Y,  it  will  be  found  by  making  .x  =  0  ;  and, 
in  general,  to  find  lohere  a  locus  crosses  or  touches  either  axis, 
make  the  other  variable  zero  in  its  equation.  Thus,  in  the  above 
example,  to  find  where  the  locus  of  y-=-  4.r  —  8  crosses  X,  make 
2/  =  0,  whence  a;=2=0/".  Making  x=0,  y  is  imaginary, 
showing  that  the  locus  does  not  meet  the  axis  of  Y.  The  dis- 
tances from  the  origin  to  the  2^oints  tchere  a  locus  meets  the  axes 
are  called  the  intercepts  of  the  locus.  They  are  distinguished  as 
the  X-intercept  and  the  5''-intercept.  Thus,  the  X-intercept  of 
7/2=  4  a;  _  8  is  0P'=2. 

7.  25?/-+ 9x-=225.  We  observe  that  the  locus  is  symmet- 
rical with  res[)ect  to  both  axes,  flaking  ?/  =  0,  we  find  x=  ±5, 
or  0^1  and  OA'  are  the  X-intercepts  ;  making  x  =  0,  y=  ±'d, 
or  OB  and  OB'  are  the  I'-intercepts.  Solving  the  equation  in 
succession  for  x  and  y,  we  have 


2/  =  ±  f  V25  -  ic2,  a-  =  ±  f  Vi)  -  y-. 

From  the  value  of  y  we  see  that  x  cannot  be  numerically  greater 
than  ±5,  otherwise  y  is  imaginary  ;  hence  no  point  of  the  curve 


THE   LINE. 


17 


lies  to  the  right  of  A  or  to  the  left  of  A' ;  that  is,  x  —  ±i)  gives 
the  limits  of  the  curve  iu  the  direction  of  X,  and  these  values 
are  the  roots  of  the  eqxiation  obtained  by  puttincj  the  quantity 
under  the  radical  sign  equal  to  zero.  The  reason  for  this  is 
plain  :  y  is  real  when  25  —  x^  is  positive,  and  imagiuar}'  when 
25  —  0."^  is  negative  ;  hence  the  limiting  values  of  y  correspond 
to  25  —  ar  =  0,  since  in  passing  through  zero  25  —  x-  changes 
sign.  For  a  like  reason,  placing  9— 2/"  =  0,  y  =  ±S  are  the 
limits  of  the  curve  in  the  direction  of  Y.  And,  iu  general, 
whenever  the  equation  of  the  locus  is  of  the  second  degree  loith 
respect  to  one  of  the  variables,  if  ice  solve  it  for  that  variable, 
and  place  the  radical  equal  to  zero,  the  roots  of  this  equation  are 


Fia.  13, 


the  limits  in  the  direction  of  the  other  axis.  (Thus,  in  Example 
6,  the  equation  is  of  the  second  degree  with  respect  to  y  ;  solved 
for  y,  the  radical  placed  equal  to  zero  gives  4  a;  —8  =0,  or  x—2. 
Beyond  this  limit  the  curve  extends  indefinitely  in  the  direction 
of  X.)  We  have  now  determined  the  intercepts,  symmetry, 
and  limits,  of  the  locus,  and  so  have  a  general  knowledge  of 
its  form  and  position.  Points  may  now  be  constructed  as 
before.     Thus,  for 

a;  =  3,  or  -  3,     y  =  ±i^,  locating  P' ,  P°,  P"',  and  P'^, 

X  =  4,  or  -  4,     y  =  ±  f ,    locating  P^',  P'\  P"S  P^,  etc. 


18 


ANALYTIC    GEOMETRY. 


8.  16^"  —  9;r- =  —  144.  Making  x  =  0,  y  is  imaginary; 
hence  the  locus  does  not  meet  the  axis  of  Y.  Making  y  =  0, 
x  =  ±i,  or  OA  and  0A\  the  X-intercepts.  The  curve  is 
symmetrical  with  respect  to  both  axes.     Solving  for  x, 


.T  =  ±  I V?/-  +  9  ; 

but  y-  +  9  cannot  change  sign,  or,  otherwise,  y-  -\-i)  =  0  gives 
imaginary  values  for  y,  hence  there  are  no  limits  in  the  direc- 
tion of  Y,  the  curve  extending  indefiuitely  in  that  direction. 
Solving  for  y, 

2/  =  ±f  Va;--16. 

Placing  a;-—  IG  =  0,  the  limits  in  the  direction  of  X  are  seen  to 
be  +  4  and  —  4.  Having  found  the  limits,  it  is  always  neces- 
sary^ to  see  whether  the  locus  lies  within  or  without  the  limits. 
In  this  case  x  cannot  be  numerically  less  than  ±  4,  and  the 
curve  therefore  lies  without  the  limits.  Having  thus  deter- 
mined the  general  features  of  the  locus,  we  proceed  to  construct 
a  few  points.     For 

x=±o,    ?/=±f, 
locating  P',  P",  P™,  P^, 

x  =  ±  6,   y  =  ±  f  V5, 
locating  P^,  P",  P'",  P^'",  etc. 

A  curve  of  this  kind,  com- 
posed of  two  separate  branches, 
Fig.  14.  is  said  to  be  discontinuous. 

9.  x- -i-y- —  Sx  — 4y  — !) —0.     Making  .t  =0,  ?/ =5,  and  —1, 

giving  the  intercepts  OB,  OB'.  For  y=0,  x=4:±  V2T=  4  ±4.6 
nearly,  or  8.G  and  —  .6  for  the  intercepts  OA,  OA'.  Solving 
for  X,  we  have 


whence 


X 


=  4±^-y'  +  Ay  +  2l. 


(1) 


THE   LINE. 


19 


Now,  every  value  of  y  gives  two  values  for  x  of  the  form 
a;  =  4  ±p,  and  thus  locates  two  points  distant  p  (the  radical) 
from  a  line  parallel  to  I''  and  4  units  from  it.  Thus,  for  y  —  (J, 
x  =  4  ±  3,  locating  P'  and  P",  each  distant  3  units  from  DD', 
DD'  being  parallel  to  Y  and  4  units  from  it.  Solving  for  y, 
we  have 

whence 


y=2  ±  V-.^•^  +  8a;+9,  (2) 

from  Avhich  we  see  the  locus  is  also  symmetrical  with  respect 
to  CC\  parallel  to  X  and  2  units  above  it ;  and,  in  general, 
whenever  the  equation,  all  its  terms  being  transposed  to  the  first 
member,  is  of  the  form  Aa?  4-  -B.«  +  etc.  ivith  respect  to  either 
variable,  if  the  coefficient  of  the  square  be  made  positive  unity, 
then  half  the  coefficient  of  the  first  poiver,  ivith  its  sign  changed, 
will  be  the  distance  from  the  other  axis  of  a  line  of  symmetry 
parallel  to  that  axis.  Thus,  v?  -\- y^ —IQy  -\- -i  —  Q  is  symmetri- 
cal with  respect  to  Y,  and  also  with  respect  to  a  line  parallel 
to  X  and  5  units  above  it;  xr -\-2x -\-y' —  ^y  =  0  is  symmet- 
rical with  respect  to  two  lines,  one  parallel  to  Y  at  a  distance 
1  to  its  left,  the  other  parallel  to  X  at  a  distance  f  above  it. 
To  find  the  limits  along  X,  put  the  radical  in  (2)  equal  to 


zero,  whence  x 


9  and  —  1. 
Values  of  y  are  imaginary  for 
a;  >  9  or  <  —  1 ,  and  the  locus 
lies  within  these  limits.  For 
the  limits  along  Y,  {\)  gives 
y  =  1  and  —  3 ,  or  no  point 
of  the  locus  lies  above  -f-  7 
or  below  —'3.  Having  now 
determined  the  intercepts, 
limits,  and  symmetry,  we 
may  construct  a  few  points. 
For 

»=  4,    ?/  =  7,    or  — 3, 
a;  =  8,   y  =  b,    or  —  1, 


\d' 

Fig.  15. 

P-, 

and  P^^, 

p\ 

and  P^'S  etc 

20  AJS"ALYTIC    GEOMETRY. 

10.  Show  that  y- —  Gy -i-x- —16  =  0,  is  symmetrical  with 
respect  to  Y,  and  a  line  parallel  to  X,  3  units  above  it ;  that  its 
limits  along  X  are  ±5,  and  along  Y,  +8  and  —2.  Determine 
its  intercepts,  and  construct. 

11.  ?/- —  10.T  +  x'^=  0.  Determine  the  lines  of  symmetry, 
intercepts,  limits,  and  construct. 

12.  af  —  6x -\-9 -j-y- -}-10y  =  0.  Lines  of  symmetry  are  —5 
and  3  from  XandT^,  respectively.  Limits  along  X  are  8  and  —2  ; 
along  Y,  0  and  —10.  Intercepts  on  Y  are  —1  and  —  9  ;  on 
X,  +  3.     Construct. 

13.  ?/--2ar  + 12a; -22  =  0.  Show  that  the  locus  has  no 
limits  in  the  direction  of  X,  lies  wholly  outside  the  limits  ±  2 
in  the  direction  of  I",  has  X  and  a  parallel  to  Y  distant  +  3 
units  from  it  for  lines  of  symmetry,  and  ±  V22  for  F-inter- 
cepts.     Construct. 

14.  y"=^9x.  This  locus  is  symmetrical  with  respect  to  X, 
is  without  limits  along  I",  has  x  =  0  for  a  limit  along  X,  lying 
wholly  in  the  first  and  fourth  angles.  Construct.  Observe 
that  if  x  =  0,  y  =  0,  and  converseh' ,  or  the  intercepts  are  zero 
on  both  axes,  and  hence  the  locus  passes  through  the  origin. 
Otherwise,  the  coordinates  of  the  origin  satisf}^  the  equation, 
and  the  origin  is  therefore  a  point  of  the  locus.  Evidently  this 
cannot  be  the  case  when  the  equation  contains  an  absolute 
term.  Hence,  in  general,  tvhenever  the  equation  of  the  locus 
contains  no  absolute  term,  the  locus  j^^^sses  through  the  origin. 
Thus,  cc^  -}-y-  —  lOy  =  0,  x*  —  y^  ■{-  3x  =  0  pass  through  the 
origin. 

15.  xy  =  10.     Solving  for  y,  y= — ■•     By  assigning  values 

to  x,  and  deriving  those  of  y,  we  may  construct  the  locus  b}' 
points.  But  the  student  should  endeavor  in  all  cases  to  de- 
termine the  general  features  of  the  locus  by  an  inspection  of 
its  equation.  In  this  instance  we  observe  that  there  is  no 
line  of  symmetry  parallel  to  either  axis,  as  the  equation  con- 
tains the  square  of  neither  variable  ;   also,  that  y  is  positive 


THE   LINE. 


21 


0 


when  X  is  positive,  and  negative  when  x  is  negative,  and  tliere- 
fore  the  curve  lies  wholly  in  the  first  and  third  angles.  Again, 
when  x  =  0,  ^  =  cc ,  and 
as  X  increases  y  dimin- 
ishes, but  becomes  zero 
only  when  .r  =  oo .  In 
the  first  angle,  then, 
the  locus  lies  as  in  the 
figure,  continnally  ap- 
proaching the  axes  as  — 
X  changes,  but  touching 
neither  within  a  finite 
distance  from  the  origin. 
A  line  to  ivldcli  a  cnrve 
thus  continually  ap- 
proaches, hilt  does  not 
touch  icithin  a  finite 
distance  Is  called  an  asymptote.  In  the  third  angle,  x  being 
negative  and  decreasing  algebraically,  y  increases  algebraically, 
becoming  zero,  liowever,  only  when  a;  =  — go.  The  axes  are 
thus  asymptotes  to  both  branches.  Constructing  a  few  points, 
we  have,  for 

x=  ±1,     y=  ±10,  P\  P", 

a;_=  ±  2,     ?/=  ±  5, 

x=  ±5,     y=  ±2, 

x^  ±  10,  .y=  ±  1, 
etc. 


Fig.  16. 


P"S  P'\ 

p\  P", 


/ 


o 


THE    POLAR    SYSTEM. 


18.  Polar  equations  of  loci.  We  have  seen  that  the  eqna- 
tion  of  a  locus  is  the  algebraic  expression  of  the  law  governing 
the  motion  of  the  point  which  traces  the  locus,  and  that  the 
quantities  in  terms  of  which  this  law  is  expressed  are  the  coor- 
dinates of  the  moving  point  and  certain  constants.  Nothing  in 
this  statement  restricts  us  to  tlie  use  of  any  particular  system  of 


22  ANALYTIC    GEOMETRY. 

coordiuates.     Thus,  if  the  law  which  controls  the  moving  point 

is  that  it  shall  always  remain  at  a  given 

distance  from  a  given  point,  the  line 

traced  will  be  a  circle.      C  being  the 

fixed  point  and  CP  =  R  the  radius  or 

constant  distance,  if  we  assume  OA  as 

the  polar  axis,  and  0  the  pole  at  a 

distance  from  C  equal  to  the  radius, 

^"^'  ^ '  ■  OP  =  r  and  A  OP  =6  will  be  the  polar 

coordiuates  of  P  the  moving  point ;  and  since  OPB  will  be  a 

right  angle  for  every  position  of   P  while  tracing  the   circle, 

OP 

— —  =  cos  BOP,      or       r=2R cos  6 

is  true  for  every  position  of  P  on  the  circle,  but  is  true  for  no 
point  within  or  without  the  circle.  It  is  therefore  the  expression 
of  the  relation  existing  between  the  coordinates  of  any  and 
every  point  of  the  circle,  and  is  therefore  the  polar  equation  of 
the  circle.  And,  conversely,  the  circle  is  the  path  of  a  point  so 
moving  that  its  polar  coordiuates  satisfy  the  above  equation  ; 
hence  the  circle  is  the  locus  of  the  equation. 

19.  Construction  of  polar  equations.  In  a  polar  equation, 
the  varial)les  which  correspond  to  x  and  y  of  the  rectilinear  sys- 
tem are  r  and  6,  and  by  assuming  values  for  one  and  deriving 
the  corresponding  values  of  the  other  from  the  equation,  we 
may  construct  as  many  points  of  tlie  locus  as  we  desire.  It  is 
obviously  convenient  to  make  ^,  the  vectorial  angle,  the  inde- 
pendent variable,  and  derive  the  values  of  r. 

Examples.  1.  r=5.  This  ecjuation  is  independent  of  0, 
that  is,  /•  =  5  =  rt  constant,  for  all  values  of  0.  It  is  tlien  evi- 
dently the  equation  of  a  circle  whose  radius  is  5,  the  pole  being 
at  tlie  centre.  We  have  seen  (Art.  14)  that  tlic  corresponding 
rectangular  equation  of  the  circle  is  or  -\-y-  =  R-.  Tlie  student 
will  observe  the  comparative  simplicity  of  tlie  polar  form  r  =  R, 
and  will  thus  see  that  in  many  cases  it  might  be  preferable  to 


THE   LINE.  23 

use  the  polar  rather  thau  the  rectangular  equation  of  a  locus 
because  of  its  simpler  form. 

2.  r=  10 cos ^.  As  in  the  case  of  rectangular  equations,  the 
student  should  endeavor  to  obtain  a  general  idea  of  the  form 
and  position  of  the  locus  from  its  equation,  rather  than  to  con- 
struct the  locus  mechanically  by  points.  In  the  present  case 
we  see  that  when  6  —  0°,  cos  $  has  its  greatest  value,  and  there- 
fore also  r ;  that  as  6  increases,  cos  0,  and  therefore  also,  ?•, 
diminishes,  becoming  zero  when  6  =  90°.  That  as  6  increases 
from  90°  to  180°,  r  is  negative  and  increasing  numerically, 
becoming  —  10  when  9  =  180°,  the  same  numerical  value  which 
it  had  for^  =  0°.     Constructing  a  few  points,  we  have,  for 

^=      0°, 

e=  oO°, 
e=  Go°, 

0=  90°, 
^=120°, 
^=l.-)0°, 
^=180°, 

As  when  6  =  {f°  the  radius  vector  coincides  with  the  polar  axis, 
P'  is  constructed  l)y  making  OP'  =  10.  Laying  off  AOP''  =  30°, 
and  OP''  =  5  V3,  P"  is  (5  VS,  30°) .  6  =  90°,  gives  r  =  0,  and 
locates  the  pole.  P'^'  and  P^'are  constructed  in  the  same  way, 
but  the  values  of  r  when  0  >  90°  being  negative  are  laid  off 
awav  from  the  end  of  the  measurinsj  arc.  If  6  increases  from 
180°  to  360°,  the  values  of  r  are  repeated  (numerically),  so  that 
the  entire  locus  is  traced  for  values  of  6  from  0°  to  180°.  As 
0P'=  10,  and  r=  lOcos^  is  true  for  all  positions  of  P,  OP^'P', 
OP^S^'i  etc.,  is  always  a  right  angle,  and  the  locus  is  therefore 
a  circle  whose  radius  is  W.  ■■^ 

The  above  loci,  and  those  of  Art.  17,  are  constructed  simply 
to  familiarize  the  student  with  the  meaning  of  the  terms  loci  of 
equations,  and,  conversely,  equations  of  loci.  A  clear  concep- 
tion of  these  terms,  and  of  a  coordinate  system  as  a  device  for 


24  ANALYTIC    GEOMETRY. 

representing  lines  by  equations,  is  fundamental  to  the  subject. 
In  Chapter  II  we  shall  begin  the  systematic  study  of  loci  by 
means  of  their  equations,  commencing  with  the  simplest,  namely, 
the  straight  line. 

20.  General  notation.  Any  equation  of  a  locus  referred  to  a 
rectilinear  system  of  axes  may  be  represented  b}'  the  equation 
f(^x,  y)  =  0,  read  '  function  x  and  y  =  0,'  this  being  a  general 
form  for  what  the  equation  of  the  locus  becomes  when  all  its 
terms  are  transferred  to  the  first  member.  In  such  an  equation, 
X  and  y  are  said  to  be  implicit  functions  of  each  other.  If  the 
equation  of  the  locus  is  solved  for  one  of  the  variables,  as  y,  the 
corresponding  general  form  will  be  y  =f{x),  read  '  y  a  function 
of  X.'  In  such  an  equation,  the  way  in  which  y  depends  upon  x 
being  fully  indicated  by  the  solution  of  the  equation,  y  is  said 
to  be  an  explicit  function  of  x.  The  primary  object  of  Algebra 
is  the  transformation  of  implicit  into  explicit  functions,  and 
/(.T,  y)  =  0  may  be  written  y  =f{x)  whenever  the  former  can 
be  solved  for  y. 

Similarly /(/•,  6)  =  0,  and  r  =f(0)y  are  general  forms  for  the 
equation  of  any  locus  referred  to  a  polar  system. 


KELATION   RECTILINEAR   AND   POLAR    SYSTEMS. 


25 


SECTION   III. 


RELATION  BETWEEN  THE   RECTILINEAR  AND   POLAR 

SYSTEMS. 

21.  Transformation  of  coordinates.  It  is  evident  that  the 
coordinates  of  a  point  and  the  form  of  the  equation  of  a  locus 
will  depend  upon  the  system  of  reference  chosen  and  its  posi- 
tion. Thus,  the  coordinates  of  P  (Fig.  19)  referred  to  the  oblique 
system  XxOiTi  are  Oi?ni  and  vHjP;  referred  to  the  rectangular 
system  XOY^  they  are  Om  and  mP\  while  if  the  polar  system 
O2A  is  employed  they  are  O^P  and  AO2P.  Again,  we  have  seen 
that  the  equation  of  a  circle  referred  to  rectangular  axes  through 
the  centre  (Fig.  20)  [sx^-\-if  =  R^  (Art.  14) ,  but  if  it  is  referred 
to  the  system  XiOiYi  its  equation  is  (x^  —m)-  +  (^1  —  ?i)-  =  R^ 

Y 


Fig.  20. 

(Ai*t.  16),  the  subscripts  being  used  to  distinguish  the  coordi- 
nates of  the  two  systems.  Again,  in  Art.  18,  we  found  the 
polar  equation  of  a  circle  to  be  r  =  2  i2  cos  0  when  the  pole  was 
on  the  circle  and  a  diameter  was  taken  for  the  polar  axis  ; 
while  the  polar  equation,  when  the  pole  was  at  the  centre,  was 
found  in  Art.  19,  Ex.  1,  to  be  t=  R. 

It  is  thus  clear  that  the  form  of  the  equation  of  any  locus  will 
vary  with  the  system  of  reference  chosen,  and,  from  the  above 


26  ANAI.YTIC   GEO:srETEY. 

illustrations,  that  one  form  ma}'  be  simpler  than  another.  It  is 
therefore  desirable  to  be  able  to  pass  from  one  system  to  another. 
This  passage  from  one  system  of  reference  to  another  is  called 
Transformation  of  coordinates. 

As  this  transformation  is  of  frequent  use,  it  is  important  that 
the  student  should  thoroughly  understand  its  object  and  nature. 
The  problem  may  be  thus  stated  :  Having  given  the  equation  of 
a  locus  referred  to  one  sj^stem  of  reference  (as  the  equation  of 
the  circle  (a;i— ???)-+  (yi  —  n)-  =  R-  referred  to  the  axes  XyOiYi)^ 
to  find  its  equation  when  referred  to  any  other  system  (as  the 
parallel  system  XOl",  to  which  when  the  same  circle  is  referred 
its  equation  is  x--\-y-  =  Br).  Tlie  object  of  this  transformation 
is  to  obtain  a  simpler  equation  of  the  same  locus  ;  the  method 
will  consist  in  finding  values  for  the  coordinates  iKj,  ?/i,  in  terms 
of  the  coordinates  x  and  y,  and  substituting  these  values  in  the 
given  equation  ;  the  resulting  equation  will  then  be  a  relation 
between  the  new  coordinates,  and  therefore  the  equation  of  the 
locus  referred  to  the  new  axes. 

In  the  same  way,  having  given  the  equation  of  a  locus  in 
terms  of  x  and  ?/,  we  pass  to  the  polar  equation  of  the  same 
locus  by  substituting  for  x  and  y  their  values  in  terms  of  r  and  9  ; 
the  resulting  equation  will  then  be  independent  of  x  and  ?/,  and, 
being  a  relation  between  r  and  0  true  for  all  points  of  the  locus, 
is  its  polar  equation.     The  problem  thus  reduces  to : 

Tlie  coordinates  of  any  point  P  with  respect  to  one  system  of 
reference  being  known,  to  find  its  coordinates  icith  resjyect  to  any 
other  system. 

The  system  to  which  the  transformation  is  made  is  called  the 
new  system  ;  that  from  which  we  pass,  the  primitive  system. 
The  three  following  cases  will  be  considered  : 

(A).  To  pass  from  any  rectilinear  system  to  any  other  recti- 
linear system. 

(J5).  To  pass  from  any  rectilinear  system  to  any  polar 
system. 

(C).  To  pass  from  any  polar  system  to  any  rectilinear 
B3'stem. 


TRANSFORMATION    OF    AXES. 


27 


RECTILINEAR    TRANSFORMATIONS. 

22.  Formuke  for  jX(ss/»(/  from  any  rectilinear  system  to 
another. 

Let  XOY  be  the  primitive  system,  ^  being  the  iucliuation  of 
the  axes,  and  P  any  point  whose  primitive  coordinates  are 
Om  =  X,  mP=y.  Let  XiOi\\  be  the  new  system,  its  position 
being    given    by    the    coordi-  ,-^-  /I'j 

nates  of  its  origin ,  OA  —  .r„, 
AOi  —  2/o»  and  the  angles  y,  yi, 
which  its  axes  make  with  the 
primitive  axis  of  X,  the  coor- 
dinates of  P  referred  to  the 
new  system  being  OiWi  =  x^ 
and  miP=  2/i-  Draw  OiB  and 
niiC  parallel  to  OX,  and  m^D 
parallel  to  0  r.     Then  Fig-2i. 

Om  =  OA  +  OiD  +  m,C. 

But     Om  =  x,  0A  =  a'o,  OyD  :  Oitn^ : :  sin  Oim^D  :  sin  OiDnii, 

whence 

Oimi  sin  Oim^D  _  Xi  sin  (^  —  y) 
^     ~       sin  OiDnii      ""         sin  /3 


and 

whence 

Substituting  these  values. 


MiC :  miP : :  sin  rUiPC :  sin  niiCP, 

2/i  sin  (^  -  yi) 
sin  /8 


CC  —  ^(j  "P 


Xi  sin  (/?  -  y)  +  yi  sin  (^  -  yQ 


sin  )8 

Again,  mP=  AOi  + Dm^+CP. 

But     ?»P=  y,  AOi  =  2/o,  I>?"i :  OiWi :  :  sin  DOim{ sin  OiDmi, 

a-,  sin  y 
whence  i^mj  =  — -. — ^ ' 

sin  13 


28 


ANALYTIC   GEOMETRY. 


and 
whence 


CP:  m^P:  :  sin  CmiP:  sin  m-^CP^ 


CP  = 


Vi  sin  yi 
sin  ji 


Substituting  these  vakies, 


y  =  yo  + 


Hence 


Xi  sin  y  -f  jji  sin  yj 
sin)8 


,  a;iSin(/3-y)  +  ?/iSin(^-yi)  .risiny+?/iSinyi  .,. 

are  the  required  formulae. 

The  following  special  cases  may  arise  : 

First.      To  pass  from  any  system  to  a  jKiraUel  one. 

In  this  case  (Fig.  22)  y  =  0,  yi  =  /S,  and  the  general  formulae 
(1)  become 

a;  =  .T„  +  .^i,  y  =  yo  +  yi,  (2) 

which  are  independent  of  /3  and  apply  to  all  parallel  axes, 
oblique  or  rectangular. 


A  "' 

Fig.  23. 

Second.     To  pass  from,  rectangular  to  oblique  axes. 

In  this  case  (Fig.  23)  ^=90°  ;  and  since  sin(90°-vl)  =  cos^l, 
sin  (90°  — y),  and  sin  (90°  — y,),  become  cos  y  and  cos  yj,  or 
the  general  formulae  become 

n-  =  a-o  +  a:i  cosy +  .'/,  cos  yi,     ?/  =  ?/„  + -''i  sin  y  +  ^/i  sin  yj.      (3) 


TRANSFORMATION    OF   AXES. 


29 


Third.      To  pass  from  one  rectangular  system  to  another. 
lu  this  case  (Fig.  24)  ^  =  90°,  y,  =  90°  +  y  ;  and  since 

sin  (90°  +  ^)  =cos^, 

sinyi  =  sin  (90°  -|-  y)  =  cosy, 

sin  {ft  —  y)  =  cosy, 

sin(/3-yi)  =sin(90°-90°-y)  =sin-y=  -siny, 

and  the  aeneral  formulae  become 


a;  =  (t'o  +  .^•lCOSy  —  2/1  siny,       y  =  y^  +  Xj  sin  y  +  t/i  cos  y . 

r 


(4) 


0     A  m 

Fis.  24. 


niA 


Fig.  25. 


Fourth.      To  pass  from  oblique  to  rectangular  axes. 

In  this  case  (Fig.  25)  y^  =  90°  +  y  ;  and  hence 

sin(/3-yi)  =  sin[/3-(90°  +  y)]  =  sin-[90°-(/3-y)] 
=  -sin  [90°  -  (^  -  y)]  =  -cos(/3  -  y), 

and  the  general  formulae  become 

aJi  sin  (/3  -  y)  -  y-^  cos  (/3  -  y)      ~] 


X  —  X(f-\- 


y  =  yo-\- 


sin^ 

Xi  sin  y  +  t/i  cos  y 
sin/3 


(5) 


The  student  will  observe  that  the  special  formulae,  like  the 
general  ones,  may  be  deduced  directly  from  the  accompanying 


figures. 


30  AISTALYTIC    GEO^METEY. 

If  the  new  origin  coincides  with  the  primitive  origin,  Xq  and 
2/0  ill  the  above  formulae  become  zero.     Hence, 

To  2MSS  from  one  oblique  set  to  another, 

X  z=  ^^  ^^^^^  -  7)  +  Ih  sip  (/?-  yO  ^  .r,  siny  +  .Visinyi       ,^. 

sin/8  "^  bin/3  ^   ^ 

To  pass  from  a  rectangular  to  an  oblique  set, 

ic  =  .Ti  cos  7  +  2/1  cos  yi,  y  =  iCi  sin  y  +  ?/i  sin  yi.     (7) 

To  pass  from  one  rectangular  set  to  another, 

X  —  Xi  cos  y  —  2/1  sin  y,  y  "=  ^i  sin  y  +  i/i  cos  y.      (8) 

To  2>ass  from  an  oblique  to  a  rectangular  set, 

^.  ^  g-i  sin  (^  -  y)  -  yi  cos (^  -  y)  ^  a;i  sin  y  +  2/1  cos  y       .g. 

sin/3  ^  sin/3  ^  ^ 

The  student  will  observe  that  none  of  the  above  formulje 
involve  higher  powers  of  the  new  than  of  the  primitive  coor- 
dinates, and  therefore  that  when  these  values  of  x  and  y  are 
substituted  in  any  equation,  the  transformed  equation  will 
always  be  of  the  same  degree  with  respect  to  the  variables  as 
the  primitive  equation  ;  that  is,  the  transformation  from  one 
rectilinear  system  to  another  affects  the  form  but  not  the  degree 
of  the  equation. 

POLAR     TRANSFORMATIONS. 

23.  Formulcp.  for  jxi.ssing  from  any  rectilinear  system  to  any 
polar  system.  Should  the  primitive  system  be  oblique,  we  may 
first  pass  to  a  rectangular  system  with  the  same  origin  by  equa- 
tions (9)  of  Art.  22  ;  the  problem  then  consists  in  passing  from 
an}'  rectangular  to  any  polar  system. 

Let  XOY  be  the  primitive  system,  and  P  any  point  whose 
coordinates  are  x  =  Om,  y  =  7nP.  Let  0,  be  the  pole,  its  coor- 
dinates being  OA  =  Xo,  AOi  =  yoi  and  let  the  polar  axis  make 
an  angle  a  with  the  primitive  axis  of  X.     Then  OiP=r,  and 


EELATIOJSr   RECTILINEAK   AND   POLAR    SYSTEMS. 


31 


0  =  AoOiP,  or  AiOiP,  according  as  the  polar  axis  lies  above 
or  below  O^X^,  drawn  parallel  to  OX,  i.e.  according  as  a  is  posi- 
tive  or  negative.  Hence,  in  general, 
XiOiP=^±a.  Now  Om=OJ^+0,Z), 
in  which 

Om  =  X,   OA  =  Xq, 

0,D=  OrPcos DO,P  =r cos  {0±  a). 
Hence  x  =  Xq  +  rcos(^±  a)  ; 


similarly, 


(1)    o^-i 


y  =  >/o+'''s\n(0±a).  ^ 

If  the  polar  axis  is  ^7ara??eZ  to  the  axis  of  X,  a  =  0,  and  the 
general  formulae  become 

X  =  Xq  +  r  cos  0,  y  =  yQ-j-r  sin  0.  (2) 

If  the  pole  coincides  ivith  the  origin,  Xq  =  y^  =  0,  and 

x  =  rcos{6±a).  y  =  rsm(0±a).  (3) 

If  the  pole  is  at  the  origin  and  the  polar  axis  coincident  ivith 
X,  a  =  0,  a^o  =  ?/o  =  0,  and 

a;  =  ?-cos^,  ,         y=rsmO.  (4) 

24.    Formulae   for  passing  from  any  polar  system   to   any 
rectilinear  system. 

From  Equations  (1)  of  Art.  23, 

X  —  x^)  =  r  COS  (^  ±  a)  ,        y  —  y^^  =  r  sin  (^  ±  a) . 

Squaring,  and  adding,  and  substituting  the  resulting  value 
of  ?*,  we  have,  since  sin''^4  +  cos-^l=  1, 

r=  V(.c-.r„)^  +  (?/-?/o)-', 


COS(^  ±  a)  = 


sin  {$  ±  a) 


.v-yo 

^{x-x,Y  +  {y-y,y    ^ 


(1) 


32  ANALYTIC    GEOMETRY. 

If  the  polar  axis  is  parallel  to  X,  a  =  0,  and 

r  =  V(a;  —  .To)-+  {y  —  2/o)^  cos ^  =  ~  ^^ 


sin  e  =  ^~-^° 

V(a;-Xo)2+(2/-?/o)'   . 
If  the  new  origin  is  at  the  pole,  o-y  =  2/0  =  0?  and 


K2) 


r  =  Va-'  +  /,    cos  (g  ±  g)  =        -^^       .    sm(g±a^=        -^       .(3) 

If  the  neio  origin  is  at  the  pole  and  the  neio  axis  of  X  coincides 
with  the  x>olar  axis,  a  =  0,  a-y  =  ?/o  =  0,  and 

r  =  Va?+y,    cos^  =  — .    sin^  = ^ (4) 

Var4-2/-  Vx^+y- 

By  means  of  Equations  (4)  we  may  pass  from  any  polar 
system  to  a  rectangular  system  with  the  origin  at  the  pole  and 
axis  of  X  coincident  with  the  polar  axis  ;  then,  by  Equations 
(3)  of  Art.  22,  to  any  oblique  system. 

Examples.  1.  Transform  v/ —  .t  —  4  =  0  (Ex.  l,Art.  17)  to 
a  new  set  of  parallel  axes,  the  new  origin  being  at  (0,  4). 

The  formulae  for  passing  from  any  rectilinear  system  to  any  parallel 
one  being  x=Xf,+  x^,  >J  =  1/0  + ^i,  in  which  3-„=0,  and  j/o=4,  the  values 
of  the  primitive  coordinates  in  terms  of  the  new  are,  in  this  case,  x  =  x^, 
y  =  4:  +  ijy  Substituting  these  values  in  the  given  equation,  we  have 
yj  —  .r,  =  0  for  the  transformed  equation.  As  the  subscripts  are  only  used 
to  distinguish  the  two  sets  of  coordinates,  they  may  be  omitted  after  the 
transformation  is  effected.  By  referring  to  Fig.  10,  Art.  17,  the  student 
will  see  that  the  new  origin  is  at  P',  a  point  of  the  locus,  and  that  there- 
fore the  transformed  equation  should  have  no  absolute  term. 

2.  Transform  ?/  +  0;  —  1  =  0  (YjX.  2,  Art.  17)  to  a  new  set  of 
parallel  axes,  the  new  origin  being  at  (1,  0).      Ans.  y  -\-x=0. 

3.  Transform  3?/  — 2a;  +  4  =  0  to  parallel  axes,  the  new 
origin  being  (  —  4,  —7).  Ans.   'Sy—2x  —  0  =  0. 

4.  Transform  ?/-  =  4.i;  — 8  (Ex.  G,  Art.  17)  to  parallel  axes, 
the  new  origin  being  at  (2,  0),  that  is,  at  P',  Fig.  12. 

Ans.  y'  =  4:X. 


RELATION   RECTILINEAR   AND    POLAR    SYSTEMS.        33 

5.  Transform  y-=ix  —  8  to  anew  set  of  rectauguhir  axes 
with  the  same  origin,  the  new  axis  of  X  making  an  angle  of 
—  90°  witli  the  primitive  axis  of  X. 

The  forinulaa  are  x  =  x\  cos  y  —  y^  sin  y,  y  —  3\  sin  y  +  i/i  cos  y,  in  which 
y  =  —  90°.  They  become,  then,  x=  i/^,  y  =  —  x^  -  Substituting  and 
omitting  subscripts,  a:^  =  4  y  —  8. 

6.  Transform  x" -\- if  ~8x  -  4.y -b  =  0  (Ex.  9,  Art.  17) 
to  a  new  set  of  parallel  axes,  the  new  origin  being  at  (4,  2), 
that  is,  at  Oi  (Fig.  15) .  Ans.  x-  +  2/"=  25. 

7.  Transform  .T^=  10  (Ex.  15,  Art.  17)  to  rectangular  axes 
with  the  same  origin,  the  new  axis  of  X  making  an  angle  of 
45°  with  the  primitive  axis  of  X. 

The  formulae  are  x  =  x^  cos  y  —  jji  sin  7,  y  =  x-^  sin  7  +  ^1  cos  7,  in  which 
7  =  4.5°,  and  they  become  x  =  Vl  (x^  —  y^),  y  =  \/|  (.r^  +  y^) .  Substituting 
these  in  xy  =  10,  and  omitting  subscripts,  .r-  —  y-  =  20. 

8.  Transform  .t-  +  ?/-  =  25  to  a  polar  system,  the  pole  being 
at  (—5,  0),  and  the  polar  axis  coincident  with  X. 

The  formulas  are  x  =  x^  +  r  cos  d,  y  =  ^o  +  '"  ^i"  ^>  wliich  for  Xq  =  —  6, 
y^  —  0,  become  x  =  —  5  +  r  cos  Q,  y  =  r  sin  6.  Substituting  these  in 
x"^  +  y'^  =  25,  we  obtain  r  =  10  cos  6. 

9.  Transform  (.r  +  ?/-)"  =(r{x-  —  y")  to  polar  coordinates,  the 
pole  being  at  the  origin  and  the  polar  axis  coincident  with  X. 

Ans.    r^  =  a^cos2  0. 

10.  Transform  the  following  equations,  the  origin  and  the 
pole  being  coincident,  as  also  the  axis  of  X  and  the  polar  axis, 
r  =  20cos^  ;  xy  =  a\  ^^^^_    _^,  ^  ^,  _  20^-  =0  ;  7-^  -  ^^- 


sin  2  e 

1 1 .  Having  the  distance  between  two  given  points  in  a  rec- 
tangular system,  d  =  Vix"  -  x')- -{-{y"  -  y')-  (Art.  7) ,  to  find 
the  polar  formula  for  the  distance,  when  the  pole  is  at  the  origin 
and  the  polar  axis  coincident  with  X. 

Substituting  x'  =  r'  cos  0',  y^  =  r'  sin  6',  x"  =  r"  cos  0",  y"  =  r"  sin  6", 

d=  V(i-"  cos  e"  —  r'  cos  e'Y  +  (?•"  sin  6"  —  r'  sin  d'Y 
=.  Vr"%cos^e"+sin^e")+r'\cos^e'+sin^e')—2r'r  '(cos 0"cos e'+sin9"sine') 
=  Vri-^+r"-^-2  r'r"  cos  id"-e'), 


34  ANALYTIC    GEOMETRY. 

wliich  is  the  formula  of  Art.  13,  which  was  there  derived  directly  from  the 
figure. 

12.    Under  the  same  conditions  find  the  polar  coordinates  of 

the  point  midway  between  two  given  points,  having  given  its 

1               r     +      x'  +  x"     y'+y" 
rectangular  coordmates ■ ,    -l — —^ — 

2  2 

^^^^     r' cos  ^'4-/-"  cos  ^"     r'sin^'  +  /-"sin^" 
2  '  2  ' 

OlS.  The  distance  between  two  points  referred  to  a  rectangu- 
lar system  is  d=  ■\/{x"  —x'y-{-  {y"  —  y')'-  Find  the  distance 
for  an  oblique  system  with  the  same  origin,  the  new  axis  of  X 
being  coincident  with  the  i)rimitive  axis  of  X,  and  the  new  axis 
of  Y  making  an  angle  ft  with  it. 

The  formulae  are  x  =  Xi cosy  +  Vi cosyj,  ?/  =  ajjsiny  +  ^/jsinyj, 
which  for  y  =  0,  yj  =  /3,  become  x  =  x^  -{-yiCosfS,  y  =  yis'mfi. 
Substituting  these  values,  and  dropping  the  subscripts, 

d=  -Vix"  -hy"  cos  13  — x'  —  y' cos  l3y-\-{y"  sin  (3  — I/' sin  fSy 
=  V[(.t"  -  x')  +  {y"  -  y')  cos^]^^  +  [(//"  -  y')  sin/3]- 


=  V(a;"  -  x'Y  +  {y"  -  ?/)-  +  2{x"  -  x')  {y"  - y')  cos/?, 
a  result  already  obtained  in  Art.  7. 


CHAPTER   II. 

EQUATION   OF   THE  FIRST  DEaREE.      THE 
STRAIGHT  LINE. 


-ooJl^JrJOO- 


SECTION   IV.  — THE   RECTILINEAR  SYSTEM. 


EQUATIONS    OF   THE    STRAIGHT   LINE. 

25.    Every  equation  of  the  first  degree  between  tivo  variables 
is  the  equation  of  a  straight  line. 

Every  such  equatiou  may  be  put  under  the  form 

Ax-{-By-\-C=0,  (1) 

in  which  A  and  B  are  the  collected  coefficients  of  x  and  y,  and  C 

is  the  sum  of  the  absolute  terms. 

Let  P',  P",  P'",  be  three  points  on 

the  locus  of   this  equation,  whose 

abscissas    in    order   of    magnitude 

are  x',  x'\  x'".     Then,  from   (1), 

their  ordinates  y',  y'\  y'",  will  also 

be    in    order   of    magnitude.      As 

these  three  points  are  on  the  locus, 

their  coordinates   must   satisfy  its 

equation  ;  hence 

Ax'+By'  +  C=0,   Ax"  +  By"+C=0,  Ax'"  +  By'"  +  C  =  0  ; 

whence,  by  subtraction, 

Aix"-  x')  +  B{y"-y')  =  0,  A  (x'"-  x')  +B{y"'-y')  =  0. 


Fig.  27. 


36 


ANALYTIC    GEOMETRY. 


Equating  the  values  of  A, 


y"'-y'. 

x"'-x' 


y"-y'. 


(2) 


Let  P'Q  be  drawn  parallel  to  OX.     Then,  from  (2), 

P"'Q^P"R 
P'Q       PR 

Hence  the  triangles  P"'QP\  P"MP',  are  similar,  and  P"  is  on 
the  straight  line  P'P".  In  the  same  manner  it  ma}'^  be  shown 
that  every  other  point  of  the  locus  is  on  the  same  straight  line 
P'P'".     The  locus  is  therefore  a  straight  line. 

The  expression  "  the  line  Ax  +  By  +  C=  0  "  will  frequently 
be  used  for  brevit}^,  meaning  "the  line  whose  equation  is 
Ax  +  By  +  C=0."  ^ 

Oblique  Axes.  The  above  demonstration  depends  only  upon  tbe  similarity  of  the 
triangles  and  is  therefore  equally  true  for  an  oblique  system. 


26.  Common  forms  of  the  equation  of  a  straight  line.  There 
are  three  common  ways  of  determining  the  position  of  a  straight 
line  3IN  with  reference  to  the  axes.     First,  by  its  intercepts 

OR,  OQ;  second,  by  its  y-intercept 
OQ,  and  the  angle  XRQ  which 
the  line  makes  with  the  axis  of  X 
(always  measured,  as  in  Trigonom- 
etry, from  OX  to  the  left)  ;  thiixl, 
by  the  length  of  the  perpendicular 
OD  let  fall  from  the  origin  on  the 
~^  line,  and  the  angle  XOD  which 
this  perpendicular  makes  with  tlie 
axis  of  X.  In  each  case  the  posi- 
tion of  the  line  is  evidently  com- 
pletely determined.  We  are  now 
to  find  the  equation  of  the  line 
when  given  in  each  of  these  three  different  ways. 

First.  Let  P  be  any  point  of  the  line,  x,  y,  its  coordinates, 
and  OR  =  a,  OQ  =  b,  the  given  intercepts.     Then 


Fig.  28. 


THE   RECTILINEAR    SYSTEM.  37 

QO  :  OR  :  :  PL  :  LR,    or    h  :  a  :  :  y  :  a  -  x, 
whence  hx-\-ay—  ab,  or.  dividing  by  a6, 

ah 

Second.  Draw  OS  parallel  to  MX,  and  let  tan  XRQ  =  m. 
Then  LP  =  SP  -  SL.     But 

SP=  OQ  =  b, 

SL  =  tanSOL .  OL  =  tan ORP  .  OL  =  -t&nXRQ .  0L=  -  mx. 

Hence  y  =  mx  -\-  b.  (2) 

The  tangent  of  the  angle  which  the  line  makes  with  the  axis 
of  X  is  called  the  slope. 

Third.  Draw  LK  parallel  to  MN,  and  PT  parallel  to  OD. 
LetXOZ)  =  a  and  OZ>=i).     Then   0K+TP=0D.     But 

0K=  OL  cosLOK^  x  cosa,    TP=  LP  sin  TLP  =  y  sina. 

Hence  x  cosu  +  y  sina  =  p.  (3) 

All  these  equations  are,  as  they  should  be,  of  the  first  degree 
(Art.  25). 

Other  forms  of  the  equation  of  a  straight  line  might  be  found 
by  assuming  other  constants  to  fix  its  position,  and  such  forms 
will  be  given  later.  The  reason  for  employing  more  than  one  is 
that  one  form  is  often  more  convenient  than  another  for  the 
solution  of  certain  problems.  Equation  (1)  is  called  the  intercept, 
Equation  (2)  the  slope,  and  Equation  (3)  the  normal  form,  while 
the  general  equation  Ax -{-By  +  (7=  0  is  called  the  general  form. 

The  student  will  observe  that  Art.  25  is  an  illustration  of  the 
general  problem  :  Given  the  equation,  to  determine  the  locus  ; 
Avhile  this  article  illustrates  the  inverse  problem  :  Given  the  law 
of  the  moving  point  {straight  line)  and  the  position  of  the  locus 
(by  the  constants),  to  determine  its  equation.  In  the  latter 
case,  the  problem  always  consists  in  finding  a  relation  between 
X  and  y  true  for  every  point  of  the  locus,  and  expressing  this 
relation  in  the  form  of  an  equation.     Whenever  we  have  sue- 


i^ 


38 


ANALYTIC   GE0:METRY. 


ceeded  in  establishing  this  equation,  we  have  the  equation  of  the 
locus,  whatever  the  constants  involved. 

Oblique  Axes.  Whatever  the  angle  A'Or(Fig.  28),  the  triangles  QOR  and  PLR 
are  similar;  hence  the  intercept  form  applies  without  change  to  oblique  axes. 

For  the  slope  form  we  have,  as  before,  LP  =  HP—  SL,  in  which  LP=y,  SP=  h;  but 
SL  :  LO  : :  sin  SOL  :  sin  LSO.  Let  u  =  XO  Y=  the  inclination  of  the  axes,  and  A=  XRQ, 
the  angle  made  by  MN  with  A".    Then 

SL  :  ar  : :  sin  A  :  siu  (A  —  <o) , 


whence  SL  =  - 


.T  i?i  n  A 


-,  and  the  equation  be- 
sln  («)  — A) 

X  +  b.   This  may  be  written 


Fig.  29. 


comes  ij= 

sin  (u)  —  A) 

in    the    form    y=mx  +  h,    understanding    that 

m=  — ^^5 .  when  the  axes  are  oblique.     If 

Bin  (<o  —  A) 

(0  =  90^,    sin  (cu  — A)  =  cos  A,   and    7H=tanA,    as 

in  (2). 

For  the  normal  form,  ROD  being  a,  as  before, 

\exr>OQ^fi.    Then 

OD  =  OK  +  7'P  =  X  cos  a  +  y  cos  /3  =i>. 

When  AOr=  90^,  DOQ  is  the  complement  of 
XOD,  that  is  of  a,  cos^  =  sin  a,  and  the  equation 
reduces  to  (3). 


27.  Derivation  of  the  common  forms  from  the  general  form. 
Since  the  equation  of  every  straight  line  i.s  of  the  general  form 
Ax-\-Bi)  +  C=0,  it  must  evidently  be  possible  to  derive  the 
common  forms  from  the  general  form,  and  to  express  the  par- 
ticular constants  a,  Z>,  ?/i,  ^>,  cosa,  sin  a,  in  terms  of  the  general 
constants  A,  B,  C. 

First.  TJie  intercept  form.  Assuming  Ax  +  By  +  C=Q, 
transposing  C  to  the  second  member,  and  dividing  the  equation 
by  —  C,  i'.c,  making  the  second  member  positive  unitv,  we  have 

X  y 


A         B 


1, 


which  is  the  required  intercept  forui,  the  intercepts  being 

a  =  -  ^    h=-^- 
A    '  b' 

Second.     The  slope  form.     Solving  the  general  form  for  y^ 
we  have  A        C 

y=  -T>^'- 


B 


B 


THE   RECTILINEAR    SYSTEM.  39 

A                   0 
which  is  the  required  slope  form,  in  which  m.  =  —  ^,  /j= 

as  before. 

Third.  The  normal  form.  This  form  requires  that  the 
second  member  {p)  should  be  positive,  as  no  convention  has 
been  made  for  the  signs  of  a  distance  except  as  that  distance  is 
laid  off  on  the  axes  ;  and  also  that  the  sum  of  the  squares  of  the 
coefficients  of  x  and  y  should  be  positive  unity,  since 

C0S"a  +  sin'a  =  1. 

Let  R  be  the  factor  which  will  transform  the  general  to  the 
normal  form.     It  nmst  fulfil  the  condition  {RA)- +  {RB)- =  \. 

Hence  R  =  —  •    Introducing  this  factor  and  transposing 

V.1-  +  B' 

C,  we  have 

Ax  Bu        _       -C 


V  A'  +  B'      V22  +  B'      VA'  4-  B' 
in  which 

A  .  B  -C 


COSa=  ,      SUl  (I  =  —  ,    p — 


V^-1-  +  B'  V^l-  +  B'  VA'  +  B' 

To  make  the  second  member  (p)  positive,  of  the  two  signs  of 
-VA--\-B^  we  must  evidently  take  the  opposite  one  of  C. 

28.  In  tlie  preceding  article  we  have  found  the  values  of  «,  &, 
711,  p,  cos  a,  and  sin  a,  in  terms  of  ^^^1,  B,  and  C.  Cut  it  is 
unnecessary  for  the  student  to  burden  his  memory  with  these 
relations.  Thus,  suppose  we  have  given  the  straight  line 
3a;  —  4^  +  10  =  0,  and  its  intercepts  are  required  ;  we  have  only 
to  put  the  equation  in  the  intercept  form,  i.e.,  transpose  the  abso- 
lute term  to  the  second  member  and  then  divide  by  —10,  giving 

-^+^=1, 

1 0     '     10 

3"  4 

and  the  intercepts  are  seen  lo  be  a  —  —  ^^,  h=i^.  A  still 
simpler  way  of  determining  the  intercepts  is  to  make  y  and  x 
successively  zero  (Art.  17,  Ex.  (>) .     Thus,  for  .);  =  0,  y=h=^^  ; 


40  ANALYTIC    GEOMETRY. 

and  for  y  =  0,  .'c  =  «  =  — y.  Again,  suppose  the  slope  is 
required.  We  then  put  the  equation  under  the  slope  form  by 
solving  it  for  ?/,  obtaining 

and  the  slope  is  seen  to  be  m  =  |,  the  y-intercept  being  b  =  J^"-, 
as  before.  Final! v,  if  the  distance  of  the  line  from  the  orig-in 
(p)  is  required,  we  put  the  equation  nnder  the  normal  form  by 
dividing  it  b}'  V-/1'  -\-  B'  =  5,  transposing  the  absolute  term  to 
the  second  member  and  changing  the  signs  throughout  to  make 
the  second  member  positive,  thus  obtaining 

the  distance  from  the  origin  to  the  line  being  2,  and  a  lying 
between  90°  and  180°  since  its  cosine  is  negative  and  sine  posi- 
tive. The  exact  value  of  a  would  be  found  from  the  tables, 
being  the  angle  whose  cosine  is  —  |,  or  sine  is  ^. 

29.   Discussion  of  the  common  forms. 

First.      The  interaqit  form.     This  form  is 

X  ,  y      .     .       ,  .  ,               (J    J           C 
-  +  -  =  1 ,  m  which  a  = ,  o  — • 

a      b  A  B 

If  a  and  b  are  both  jwsitive,  the  line  occupies  the  position 
M^N^  (Fig.  30) ,  both  intercepts  being  laid  off  in  the  positive 
directions  of  the  axes.  If  a  and  b  are  both  negative,  the  line 
occupies  the  position  J/"iV",  botli  intercepts  being  laid  off  in 
the  negative  directions  of  the  axes.  In  like  manner  when  a  is 
positive  and  b  negative,  the  line  lies  as  does  il/"'jV'",  and  «7ie?i 
a  is  negative  and  b  jyositive,  as  does  J/'^-Y'^'.  We  observe, 
also,  that  when  C  and  B,  as  also  C  and  A,  of  the  general  form 
have  like  signs,  the  intercepts  are  negative,  and  when  they 
have  unlike  signs  tlie  intercepts  are  positive.  Ifa  =  <x,  the 
equation  becomes  y  =  b,  and  since  y  is  b  for  nil  values  of  x, 
that  is,  since  the  ecjuatiou  is  independent  of  .r,  >/  =  b  is  the 
equation  of  a  parallel  to  X  at  a  distance  b  from  it,  above  or 
below  according  as  b  is  positive  or  negative.     Notice  that  when 


THE   RECTILINEAR    SYSTEM. 


41 


a  =  00,  -4  =  0,  and  the  geueral  form  is  independent  of  x.  Sim- 
ilarly, if  b  =  <Ki,  we  have  x  =  a,  the  equation  of  a  parallel  to  Y, 
its  position  to  the  right  or  left  of  Y  depending  upon  the  sign 
of  a  ;  in  this  case  5  =  0,  and  the  general  form  is  independent 
of  y.  If  a=0,  the  line  passes  through  the  origin,  therefore 
b  is  also  zero.  In  this  case  the  intercept  form  is  inapplicable 
because  there  are  no  intercepts,  but  we  see  from  the  values  of 
a  and  b  that  (7=0,  and  the  general  form  becomes  Ax  +  J5?/  =  0, 
as  it  should,  since  Avhen  a  locus  passes  through  the  origin  its 
equation  has  no  absolute  term  (Art.  17,  Ex.  14). 


Second.    The  slope  form. 

y  =  mx  -[-  ?>,  in  which  in. 


This  form  is 

B'  B 

If  m  is  positive,  the  line  makes  an  acute  angle  with  X  and 
cuts  Y  above  or  below  the  origin  according  as  b  is  positive  or 
negative.  If  m  is  negative,  the  line  makes  an  obtuse  angle 
with  X.     We  thus  have 

y  =  —  mx  +  h,  JPN', 

y  =  -mx-b,  2P'N'\ 

y=      mx-b,  J/'"^"S 

y=     mx  +  b,  M'^N'-". 

If  m  =  0,  the  line  is  parallel 

to  X,  and  y  =  6  is  its  equation, 

as  already  seen.      If  m  —  co, 

the  line  must  be  parallel  to  Y, 

since  the  angle  whose  tangent 

is  00   is    90°.      The    equation 

then  becomes  ?/  =  x  .r  +  6.   To 

interpret  this  form,  we  observe  that,  as  the  line  is  parallel  to 

A    ,  C   „ 

the  con- 


Fig.  30. 


Y,  b  must  also  be  x ,  and  hence  in  m  =  —  ^,  6  = 

B 


B' 


The 


ditions  m=  x,  &  =  x,  will  both  be  fulfilled  when  J3  =  0 

(J 
general   form    then  becomes   Ax+C='d,  or  x  = 7  =  «r  as 

before.     If  b  =  0,  we  have  y  =  mx,  or  Ax  -]-  By  =  0,  as  before^ 


p: 


42  ANALYTIC    GEOMETRY. 

the  line  passiug  through  the  origin.  The  form  y  =  mx  is  the 
most  convenient  for  lines  passing  tlirough  the  origin,  the  value 
of  m  fixing  the  inclination  of  the  line  to  X. 

Tuna).      Tlie  7iormal  form.     This  form  is  '^ ,    <^, 


a;  cos  a  -\-y  sin  a  =j>, 
in  Nvhich 

A  .  B  C 

cos  a  =  —  ,  sin  a  = .,  2'>  ^  —  — 


V-42  +  B-  V^"  +  B'  -v/^-  +  B' 


the  sign  of  V^-l-  +  B-  being  such  as  to  make  p  positive.  If 
both  cosa  and  sina  are  positive,  a  lies  between  0^  and  90° 
(3/'iV').  If  both  are  negative,  a  lies  between  180°  and  270° 
( J/"iV^') .  If  cos  a  is  2)ositive  and  sin  a  is  negative,  a  lies  between 
270°  and  360°  (7W'"iV"'),  and  ifcosais  negative  and  siiia  posi- 
tive, between  90°  and  180°  {J^P^'N^').  If  p  =  0,  the  line  passes 
through  the  origin,  and  its  inclination  is  known  when  sina  and 
cos  a.  are  known,  its  equation  taking  the  form  x  cosa  +  y  sina  =  0, 

or        — 3r=z=r  H =  0,  or  Ax  -j-By  —0,  as  .before. 

VA'  +  B-      VA'  +  B' 

If  a  =  0°  or  180°,  sina=0,  and  the  equation  becomes 

cosa  V^  +  #         ^  ^ 

as  before,  the  line   being  parallel   to    1'.     //'  a  =  90°  or  270°, 

cosa=0,  and   y  =  —^ — =h,    in    like    manner,    the   line    being 

sin  a 

parallel  to  X. 

OnLHiL'E  Axes.     Tlie  intercept  form   beiug  the  same,  tlie  discussion  above  given 
applies  equally  to  oblique  axes. 

The  slope  form  is  ?/= ' x  +  b  (Art.  26).     Since  sin  A  is  alwaj's  positive,  the 

sin(uj— A) 

Bign  of  the  coefl'icient  of  x  clcpomls  upon  that  of  8in(w  — A),  anil  will  be  positive  or  nega- 
tive as  (u>  or  <A.     Hence  »/=  •:  — 511! ,r  i  6  represents  four  lines  situi'toii,  relative 

6in(u)  — A) 

to  the  axes,  as  y  =  i  m.r  h  b  in  the  <ase  of  rectangular  axes. 

The  discussion  of  the  normal   form  x  cosa +(/ cos ^  =  y>  (Art. '2n)   is  similar  to  the 

1}  7} 

above,  the  equation  taking  the  forms  x  =     ' — ,   y  =  — i — ,  when  the  lino  is  i>arallel  to  the 

*  "^  cosa      •'        C08/S  ' 

fczes,  i.e.,  when  ^  =  9(P  and  o=00\  respectively. 


THE    RECTILINEAIl    SYSTEM. 


43 


30.  To  construct  a  straight  line,  having  given  its  equation. 
If  the  equation  is  given  in  one  of  the  three  common  forms,  we 
may  construct  the  line  b}'  means  of  the  given  constants.  For 
example, 

■       ■■  (1) 

(2) 
(3) 


8      4 


—   1_2 
~5"' 


are  the  intercept,  slope,  and  normal  forms,  respectively,  of 
4a;  +  3y—  12  =  0.  From  the  first,  make  OR  =  3,  OQ  =  A,  and 
QR  is  the  line.  From  the  second,  make 
OQ  =  4  and  draw  QR-,  making  an  angle 


with   X,   whose  tangent  is 


4 
3' 


This 


angle  may   be  constructed  without  the 

tables    by    making    QN=2>,     NP'  =  A, 

these  lines   being  parallel  to  the   axes, 

laying  off  QN  to  the  right  or  left  of  Q, 

as  the  angle  is  acute  or  obtuse,  j.e.,  as 

wi  is  plus  or  minus.     From  the  third  lay 

off  XOD  =  angle  whose  cosine  is  |-  (or  sine  f ) ,  make  OD  =  ^, 

and  draw  QR  perpendicular  to  OD. 

Since  the  line  is  a  straight  line,  it  ma}'  evidently  be  con- 
structed by  constructing  an}'  two  of  its  points.  Thus,  for 
x  =  -3,  ?/  =  8,  (P'),  and  for  x=G,  y  =  -4,  (P").  But  the 
points  most  easily  constructed  are  those  in  which  the  line 
crosses  the  axes.  Thus,  in  any  of  the  above  forms,  for  x=  0, 
2/ =  4,  (Q),  and  for  y=0,  ic  =  3,  (R).  Hence,  practically, 
whatever  the  form  in  which  the  equation  is  given,  to  construct  a 
straight  line  from  its  equation.,  construct  its  intersections  with  the 
axes. 

Oblique  Axes.  To  construct  the  line,  make  x  and  y  in  succession  equal  to  zero,  and 
determine  the  intercepts. 

Examples.     1.  Construct  the  line  whose  equation  is  x—y=2. 
Making  y  =  0,  we  have  x  =  2,  the  intercept  on  X;  making  x  =  0,  we 


have  ?/  =  —  2,  the  intercept  on  Y. 
is  the  required  line. 


A  line  through  the  points  thus  found 


44  ANALYTIC    GEOMETRY. 

2.  Construct  the  following  lines  :  ^  —  2.i'  +  G  =  0;  x  -\- y  =  7  ; 
Sy  +  x  — [)  =  {);  3a;  +  o^  +  15  =  0. 

3.  Construct  the  line  //  +  2  x  =  0. 

Since  this  equation  has  no  absohite  term  tlie  line  passes  through  thu 
origin.  In  such  a  case  construct  any  second  point;  thus  .r  =  1  gives 
y  =  —  2.     Then  join  (1,  —  2)  with  the  origin. 

4.  Construct  the  lines  y  =  x;  y  =  —  x;  2y  —  ox=0. 

5.  AVhat  angle  does  y  +  Sx—l  —  0  make  with  X ? 

Putting  the  equation  under  tlie  slope  form,  ^  =  —  3 .(  +  7,  and  the  angle 
is  the  angle  wliose  tangent  is  —3. 

6.  What  is  the  distance  of  6  a;  +  8 .?/  +  11  =  i)  from  the  origin  ? 

Dividing  by  —  V^"^  +  B^  =  —  10,  \vc  liave  —  -i x  —  ^y  =  \\,  and  ;j  =  ii  = 
the  required  distance. 

7.  Find  the  intercepts,  slope,  distance  from  the  origin,  and 
angle  made  by  the  perpendicular  from  the  origin  on  the  line 
with  X,  of  the  line  '2x+ly  —  'd  =  0. 

Making  x~0,  y  =  b=^;  making  y  =  0,  x  =  a=?^.  Solving  for  y, 
y  =  —  'i-x+  2,  .-.  ?u  —  —  ?.     The  normal  form  is 

2x    .    7i/          9                    9  ,  -17 
1 ■— = ,  .•./;  = ,     and     a  =  sni  ^ — — 

\/53      \/53      \/53  v'SS  VSS 

8.  Find  a,  6,  m,  jh  and  a  in  the  following  lines  : 

3 >/  —  4 a;  +  25  =  0  ;   7.i-  —  //  =  0  ;  y-\-x  —  o  =  0. 

9.  The  intercepts  of  a  line  are  6,  3  ;  write  its  equation. 

Ans.  x  +  2?/  —  G  =  0. 

10.  A  line  makes  an  angle  of  45°  with  X,  and  cuts  i'at  —  2 
from  the  origin  ;  write  its  equation.  Ans.  y  —  x—  2. 

11.  The  distance  of  a  line  from  the  origin  is  (!,  and  the  per- 
pendicular upon  it  from  the  origin  makes  an  angle  of  G0°  with 
X;  write  the  equation  of  the  line.  Ans.   V3y  +  .'c=12. 

12.  Same  as  Ex.  11.  when  the  angle  is  120°. 

13.  AVrite  the  equations  of  p;u:illels  to  A",  one  4  above,  and 
one  10  below  it. 


THE    RECTILINEAIl    SYSTEM.  45 

14.  Write  Ihe  equations  of  the  sides  and  diagonals  of  a 
square  whose  side  is  10,  the  sides  being  parallel  to  the  axes 
and  the  centre  at  the  origin. 

15.  What  are  the  equations  of  the  axes? 

Since  x  =  a  is  a  parallel  to  Y  at  a  distance  =  a,  if  a  =  0  the  line  coin- 
cides with  y.  Hence  t  =  0  is  the  equation  of  Y.  Similarly  y  =  0  is  the 
equation  of  X. 

16.  Determine  which  of  the  points  (2,  3),  (1,  —3), 
(_2,  7),   (  —  3;  11)   are  on  the  line  2i/ +  7.i-— 1  =  0. 

17.  Find  the  length  of  the  portion  of  the  line  4a; +  3?/ =  24 
included  between  the  axes.  Ans.  10. 

r^     18.  The  line  y  =  mx  passes  through  {x\  y')\  find  the  value 
of  m, 

31.  Equation  of  a  straight  live  x>assing  through  a  given  point. 

Let  {x\  y')  be  the  given  point.  Since  the  required  line  is  a 
straight  line,  its  equation  will  be  of  the  form  (1)  Ax-\-By  +  C=  0, 
and  since  it  passes  through  the  point  (x\  y'),  we  have 
Ax'+By'  +  C=0.     Subtracting  this  from  (1), 

A{x-x')+B{y-y')=0.  ..,y-y'=-^{x-x').     But  -|=m. 

Hence  V  —  y'  =  "^  {^  —  ^')  •>  (2) 

being  a  relation  between  x  and  y  in  terms  of  the  given  constants 
x\  y',  is  the  required  equation. 

An  infinite  number  of  lines  may  be  drawn  through  a  given 
point ;  hence  the  line  is  not  determined  unless  its  slope  m  is  also 
given.  Thus,  the  line  through  (1,  —4),  making  an  angle  of 
45°  with  X,  is  y-\-i=l(x  —  l),  or  y  =  x—o. 

Oblique  Axes.    Th2  above  equation  applies  to  oblique  axes,  understanding  that 

„,=_jiE^ — 

f*/  sin  (u>  —  A) 

'^  32.  Equation  of  a  straight   line  passing  through   two  given 
points. 

Let  (x',y'),   {x",y")   be   the   given   points.     The   required 


46  ANALYTIC    GEOMETRY. 

equation  will  be  of  the  form  (1)  Ax  +  Bj/ -\- C=0,  since  the 
Hue  is  a  straight  line,  and  must  be  satisfied  for  the  co- 
ordinates of  the  given  points;  hence  (2)  Ax' -\- By' -\- C  =0, 
(3)  A'c"+%"+C=0.  Subtracting  (2)  from  (1),  and  (3) 
from  (2),  we  have 

A{x-x')  +  B{y-y')  =  0,   A{x'-x")-\- B (y'-y")=0. 

Transposing,  and  dividing, 

which    is  a  relation    between  x  and  y  in    terms  of   the   given 

constants,  and  hence  the  equation  required,  the  coefBcient  of  x, 

v'—  v" 

^^ — ^,  being   the    slope   (Art.  27).      Thus    the   line   passing 

through    (1,    2)    and    (-3,  4)     is    y  -  •>  =  ^^^  (x-l),     or 

1+3 

2y  +  x—b  =  0.      It   is   immaterial  which  point   is   designated 

4  —  2 
as   (x',y').     Thus,    y  -  4  =  — ^^— (.-«  + 3),  or  2?/ +  .^- -  5  =  0, 

as  before. 

Oblique  Axes.    Nothing  in  the  above  reasoning  being  dependent  upon  the  inclina- 
tion of  the  axes,  the  equation  is  the  same  if  the  axes  are  oblique,  only  the  coellicient 

•^  ~y    is  then  the  ratio  of  the  sines  of  the  angles  which  the  line  makes  with  X  and  Y. 
x'-x" 

Examples.     1.  "Write  tlie  equation  of  a  line  through  (2,  4) 
having  the  slope  5.  Ans.  y  —  5  .^•  -j-  6  =  0. 

2.  "Write  the  equation  of  a  line  through  (2,  3)  and  (1,  —2). 

In  place  of  using  Eq.  (4),  Art.  32,  as  there  illustrated,  it  is  quite  as 

expeditious  to  determine  the  constants  of  any  one  of  the  three  common 

forms  directly.     TIius,  the  form  ij  —  mx  +  h,  satisfied  in  succession  for  the 

two  points  gives 

3  =  2  »«  +  h,     and     —  2  -  m  +  h. 

Subtracting,  we  obtain  m  =  5.     Substituting  this  value  of  m  in  either  of 

the  above,  we  find  b  =  —  7.     Hence  y  =  5  .r  —  7. 

3.  Find  the  equations  of   the    sides  of   the    triangle    whose 
vertices  are  (4,8),  (1,  4),  ( — 1,  —8). 

Ans.  Sy  —  4x-H  =  0;  T)?/ -  12.x-- 8  =  0  ;  y-2x  =  0. 


THE   EECTILINEAll    SYSTEM. 


4r 


4.  Find  the  equations  of  the  raedials  of  the  triangle  of  Ex.  3, 
Ans.   ll?/-2U;c-8  =  0;  y-4x=0;  13y  -  28x- -  8  =  0. 

5.  AVrite  the  equation  of  a  line  through  (2,  5)  and  the  origin. 

We  may  use  the  equation  of  a  line  tlirough  two  points,  making  one  of 
the  points  (0,  0)  ;  or  tlie  slope  form  */  =  mx  (since  b=  0),  which,  satisfieci 
for  (2,  5),  gives  m  =  5,  and  therefore  y  =  |  x. 

6.  Write  the  equations  of  tlie  following  lines  : 

(1)  through  (  —  7,  1),  making  an  angle  45°  with  X. 

(2)  through  (2,  -1),  and  (-3,  4). 

(3)  through  (  —  1,  —7),  and  the  origin. 

(4)  through  (-G,  -3),  parallel  to  X. 

(5)  through  (  —  1,  2),  parallel  to  Y. 

Ans.  y  —  x  —  8  =  0  ;   >/  +  x  — 'i  =  0  ;  y—7x;  ?/  =  —  3  ;  x  =  — 1. 


PLANE   ANGLES. 

33.    To  find  the  angle  included  bettvemi  tivo  given  straight  lines^ 

The  slope  form  is  best  adapted  to  this  problem. 

Let  y  =  mx  +  6,  y  =  m'x  -\-  b',  be  the  two  given  lines  ;  m  and 
m'  ai-e  the  tangents  of  the  angles  XRP=X,  and  XQP  =  X\ 
Then,  if  c  =  tan  RPQ  =  tany,  from  Trig- 
onometry we  have 

tan  A'  —  tan  A 


tany= 


or 


c  = 


1  +  tan  A  tan  A' 

m'  —  7n 
1  +  mm' 


(1) 


Thus  the    tangent  of   the   angle  between 


Fig.  32. 


,  and  the  angle  may 


A   O 

y=  4.T  +  7  and  ?/=  2a;  — 1  is  c  = —    ,  ......  .^v.  ....j,- 

be  found  from  a  table  of  natural  tangents.     It  is  immaterial 
whether  we  substitute  4  for  m'  and  2  for  m,  or  vice  versa;  in 

the  latter  case  c  = = ,  the  difference  in  sign  being  due 

to  the  fact  that  the  tangents  of  the  supplementary  angles  EPQ 


48 


ANALYTIC    GEOMETRY. 


and  QPS  which  the  Hues  make  with  each  other  are  numerically 
equal  with  opposite  signs.  We  thus  obtain  the  acute  or  obtuse 
angle,  according  as  the  sign  of  the  result  is  positive  or  negative. 

34.    To  find  the  equation  of  a  straight  line  mcikincj  a  given 
angle  v:ith  a  given  line. 

Let  y  =  mx  +  h  be  the  given  line,  y  =  m'x  +  h'  the  required 
line,  and  c  =  tangent  of  the  given  angle.     Then  in  the  relation 


c  = 


m'  —  m 
1  +  '^nm'' 


c  and  m  are   known.      Solving  for  m',  we  have 


,      m  -4-  c 
m'= 


1  —  mc 


Hence  the  requu'ed  equation  is 

m  4-c 


y  = 


x  +  b'. 


(1) 


1  —  mc 

Since  an  infinite  number  of  straight  lines  may  be  drawn 
making  a  given  angle  with  a  given  line,  b'  is  undetermined. 
We  are  then  at  libertN'  to  impose  another  condition  upon  the 
line,  as  that  it  shall  pass  through  a  given  point.  The  equation 
of  a  line  through  a  given  point  is  y  —  y'  —  m'  (x  —  x') ,  in  which 
m'  may  have  any  value  (Art.  31).  Substituting  the  value 
found  above, 


y-y 


m  +  c 


{X  -  X') 


(2) 


1—  mc 

is  the  equation  of  a  straight  line  passing  through  a  given  2wint  and 
making  a  given  angle  loith  a  given  line.      Thus,  the  line  through 

(2,  4),  making  an  angle  45°  with  ?/=  2.^—4, 

\y  is2/-4=^±i(.r-2),   or   ?/  =  -3.v+10. 


Fig.  33. 


1-2  ^ 

Constructing,  MN  is  the  given  line 
?/  =  2.^  —  4;  P'  the  given  point  (2,4); 
and  P'Q  the  hue  ?/  =  -3.x'+10.  The 
student  will  observe  that  P'R  makes 
:in  angle  135°  with  MN,  the  angle  be- 
ing measured,  as  always,  from  the  line 
to  the  left ;  and  that,  therefore,  to 
obtain  the  equation  of  P'R  wo  should 
make  c  =  tan  135°  =  —  1 . 


THE   RECTILINEAR    SYSTEM.  49 

35.    Conditions  that  two  lines  shall  he  parallel,,  or  perpendicu- 
lar,, to  each  other. 

First.     If  two  lines  are  parallel,,  their  included  angle  is  zero. 

All  '     /JIT 

Hence  (Art.  33)  c= =  0,  or  m==  m' ;  th;it  is,  tivo  lines 

1+mm 

are  paralld   whenever,   their  equations  being  solved  fur  y.  the 

coefficients  of  x  are  equal.     This  follows  obviously  from  the  fact 

that  parallel  lines  make  equal  anoles  with  X,  and   hence   the 

tangents  of  these  angles,  m,m\  must  be  equal.     Thus, 

?/=2a;  +  4,  y  =  2x  —  7,  y  —  2x  =  0,  are  all  parallels. 

Cor.     The  equation  of  a  line  passing  through  a  given  point 
(x',  y')  jyarallel  to  a  given  line  y  =  mx  +  b,  is  (Art.  31) 

y-y'  ^m  {x-x').  (1) 

Second.     If  the  lines  are  perp>endicular  to  each  other,  their 

included  angle  is  90°,  and  hence  c-  = =  oo, 

1  +  7nm' 

or  l+mm'  =  0,  .•.m'= ; 

m 

that  is,  tivo  lines  are  perpendicular  to  each  other  ivhenever,  their 
equations  being  solved  for  y,  the  coeffi^cients  of  x  are  negative  recip- 
rocals of  each  other .     Thus, 

are  all  perpendicular  to     y  =      f  a;  +  7. 

Cor.     The  equation  of  a  line  passing  through  a  given  point 
(a-',  y')  perpendicular  to  a  given  line  y  =  mx  +  b 

is  y-y'= {x-x').  (2) 

m 

The  equations  m  =  m\  ra'  = ,  are  not  the  equations  of 

m 

lines,  for  they  contain  no  variables.  Since  they  involve  only 
constants  (which  serve  to  fix  the  position  of  the  lines  in  ques- 
tion) ,  they  express  conditions  imposed  upon  the  position  of  the 
lines.     Such  equations  are  called  equations  of  condition. 


60  ANALYTIC    GEOMETRY. 

Examples.  1.  Find  the  angles  between  the  lines  ?/= —a;  +  2, 
?/  =  3«  +  7;  y  =  ^x—l,  ?/  =  -f-a;  +  4;  y  =  2x-o,  i/=2x+l; 
?/  =  i  a;  -  3,  y  =  -2x  +  0:  3  ?/  +  4a-  +  1  =  0,  2.v  +  .<;  +  5  =  0. 

Ans.  c=2;  c  = -.^  ;  0°;   90°;  c  =  i.^ 

2.  Write  the  equation  of  a  line  making  an  angle  whose  tangent 
is  3  with  ?/  =  —  3x  +  4.  Ans.  y  =  6. 

3.  Write  the  equations  of  lines  making  angles  of  45°  and  135° 
with  2y  —  x  +  d  =  Q.  J^y^s_  y  =  Zx  +  h\  y  =  —  \x-^h. 

4.  Write  the  equation  of  a  line  through  (—  3,  7)  making  an 
angle  whose  tangent  is  V3  with  2y  —  a;  +  1  =  0. 

Ans.    (2  -  V3)  y  -  (1  +  2  V3)  a;  -  17  +  V3  =  0. 

5.  Write  the  equations  of  two  parallels  to  y  =  -|  x-\-l ',  also 
io'dy-\-lx  =  Q. 

^      6.    Write  the  equation  of  a  parallel  to  3?/  — 4  a.' =2  through 
(1,2).  Ans.  Zy- 4.x -2  =  0. 

7.  Write  the  equations  of  two  perpendiculars  to 

y  =  —  hx  -\-  4  ;  also  to  y  —  x  +  4  =  0. 

8.  Write  the  equation  of  a  line  through  (7,-1)  perpen- 
dicular to  ?/  =  —  4a;  +  1  ;  also  through  (7,  —  1)  perpendicular  to 
dy-2x  =  0.  Aiis.  4iy-a;  +  ll  =  0;  22/  +  3.r- 19  =  0. 

9.  Write  the  equations  of  lines  through  (1,  3)  making  angles 
of  0°,  90°,  and  45°  with  X.       jins.  y  = 'i  ;  x=l;  y-  x-2  =  0. 

10.  Write  the  equation  of  a  line  through  (5,  3)  parallel  to 
the  line  whose  intercepts  are  3,  2.  ^ns,  3?/  +  2  a;  —  19  =  0. 

11.  Write  the  equation  of  a  line  through  (2,  3)  perpendicular 
to  the  line  joining  (2,  1)  with  (—2,5).  Ans.  yz=x  +  l. 

12.  The  vertices  of  a  triangle  are  (—1,-1),  (—  3,  5), (7, 11). 
Write  the  equations  of  its  altitudes. 

yins.  3?/ -.0-26  =  0;  3?/ +  5.^-  + 8  =  0  ;  3// +  2.«;- 9  =  0. 


1 


THE   IlECTILlNEAll    SYSTEM.  51 

13.  Write  the  equations  of  the  perpeadiculais  erected  at  the 
middle  points  of  the  sides  of  tlie  triangle  of  Ex.  12. 

Ans.  3y-x-S  =  0;  3  ?/ +  o.u  -  34  =  0  ;  oy  +  2x  —  2i=0. 

14.  Prove  that  Ax  +  B>i  -j-  C  =  0  is  perpendicnlar  to 

A'x  +  B'y  +  C  =  0  if  AA'  +  BB'  =  0. 

1.0.    Prove  that  Ax  +  By  -f  C  =  0  is  parallel  to 
A'x  +  B'y  +  C  =  0  if  AB'  -  A'B  =  0. 

16   Prove  that  the  angle  between  Ax  +  By  -f-  C'=  0  and 

A'x  +  B'y  +  C  =  0  is  given  by  the  relation  tan  y  = 

-T-      J-r  to  J  ;'      AA'  +  BB' 

17.  Write  the  equation  of  a  straight  line  perpendicular  to 
Ax-\-By+  C=0  and  making  an  intercept  a  on  the  axis  of  X. 

18.  Write  the  equation  of  a  line  perpendicular  to  y  =  mx  +  b 
and  at  a  distance  d  from  the  origin. 

19.  Write  the  equation  of  a  line  parallel  to  y  =  mx  -\-h  and 
at  a  distance  d  from  the  origin. 

20.  Prove  that  if  the  equations  of  two  straight  lines  differ 
only  in  their  absolute  terms,  the  lines  are  parallel. 


INTERSECTIONS. 


/  J 


36.  Intersection  of  loci.  The  point  of  intersection  of  two 
straight  lines  is  the  point  common  to  both.  But  if  a  point  lies 
on  a  given  straight  line,  its  coordinates  must  satisfy  the  equa- 
tion of  the  line  ;  hence  the  coordinates  of  the  point  of  intersection 
must  satisfy  the  equations  of  each  line.  Conversely,  to  find  the 
point  of  intersection  of  two  straight  lines,  combine  their  equations 
and  find  the  set  of  values  of  the  coordinates  ichich  satisfies  them  both. 

The  above  reasoning  is  obviously  entirely  general.  AVhatever 
the  loci  under  consideration,  if  the}'  have  a  common  point,  or 
points,  the  coordinates  of  these  points  must  satisfy  both  equa- 
tions. Hence,  in  general,  to  find  the  intersections  of  any  two 
loci,  combine  their  equations. 


52  ANALYTIC    GEOMETRY. 

Since  the  number  of  sets  of  values  of  x  and  ?/,  obtained  by 
making  the  equations  simultaneous,  is  equal  to  the  product  of 
the  numbers  indicating  the  degrees  of  the  equations,  this  prod- 
uct also  indicates  the  possible  number  of  intersections.  If,  for 
example,  the  equations  are  of  the  second  degree,  their  loci  may 
intersect  in  four  points,  but  no  more  ;  and  as  some  of  the  val- 
ues of  X  and  y  thus  obtained  may  be  imaginary,  the  number  of 
real  intersections  may  be  less  than  four.  And,  in  general,  the 
greatest  possible  number  of  intersections  of  two  loci  whose  equa- 
tions are  of  the  j>th  and  qili  degrees,  respectively,  will  be  pq, 
and  the  number  of  real  intersections  will  be  the  number  of  sets 
of  coordinates,  satisfying  both  equations,  in  which  x  and  y  are 
both  real. 

Since  all  equations  of  straight  lines  are  of  the  first  degree, 
but  one  set  of  values  of  x  and  y  can  be  found  satisfying  any  two 
such  equations,  or  two  straight  lines  can  intersect  in  but  one 
point.  If  two  straight  lines  are  parallel,  they  cannot  intersect, 
and  the  combination  of  their  equations  will  give  an  impossible 
result.  Thus,  x-\-y  =  4:  and  x  +  y=7  are  parallels.  Com- 
bining, we  obtain  3  =  0.,  Hence  non-intersection  is  shown  by 
the  occurrence  of  impossible  or  imaginary  results.  Otherwise, 
equating  the  values  of  .r,  0?/  =  3,  or  7/  =  |-=oo,  showing  that 
the  lines  intersect  only  at  an  infinite  distance. 

Examples.     Find  the  intersection  of  the  following  lines  : 

1.  2?/  — 3a;- 7  =  0,  and  2i/+ .T- 10  =  0.         Ans.   (|,-V-)' 

2.  .T-f-2?/  — r)  =  0,  aud2.r +  //  — 7  =  0.  Ans.    (3,1). 

3.  2/  —  .r  4- 1  =  0,  and  y  -f  x  -f  1  =  0.  Ans.    (0,  -  I ) . 

4.  6^0;  -t-  6  y  —  1  =  0,  and  x-\-y  =  4. 

5.  iB  +  y  =0,  and  .^'— ?/ =  0. 

6.  Find  the  vertices  of  the  triangle  whose  sides  are 

5y-12a;-8  =  0,    3?/ -  4.x- -  8  =  0,    y-'2x  =  0. 

Ans.   (1,4),  (-4,  -S),  (4,8). 


THE    RECTILlNEAIt    SYSTEM.  53 

7.  Show  that  y  +  3x-  -  1  =  0,  ?/  +  2;i;  +  7  =  0,  y  -  .r+  31  =  0, 
meet  in  a  poiut. 

8.  Show  that  the  medials  of  the  triaugle  of  Ex.  4,  Art.  32, 
meet  iu  a  poiut. 

9.  Show  that  the  altitudes  of  the  triangle  of  Ex.  12,  Art.  35, 
meet  in  a  point. 

10.  Show  that  the  perpendiculars  erected  at  the  middle 
points  of  the  sides  of  the  triangle  of  Ex.  13,  Art.  35,  meet  in  a 
point. 

Examples  on  the  intersection  of  curves  are  reserved  until 
the  student  is  familiar  with  the  equations  of  the  curves ;  but  he 
will  observe  that  the  process  is  the  same,  whatever  the  degree 
of  the  equations  or  the  system  of  reference  :  to  find  the  intersec- 
tions of  any  lines,  combine  their  equations. 

37.    Lines  through  the  intersection  of  loci. 

Let  (1 )  Ax  +  By  +  C  =  0,  (2)  A'x  +  B'y  +  C  =  0,  be  the 
equations  of  any  two  straight  lines,  aud  7i  any  arbitrary  con- 
stant;  tlieu  is  (3)  Ax  +  By  +  C  +  k  {A'x -j- B'y  +  C')  =  0  the 
equation  of  a  straight  line  through  their  intersection.  For  the 
values  of  x  and  y  which  satisfy  (1)  aud  (2),  evidently  satisfy 
(3)  also,  hence  (3)  passes  through  the  poiut  of  intersection  of 
(1)  and  (2).  Moreover  (3)  is  of  the  first  degree,  hence  the 
equation  of  a  straight  line. 

Note.  This  reasoning  is  entirely  independent  of  the  form  and  the 
degree  of  the  equations.  Hence  if  a  =  0,  ;3  =  0,  be  the  equations  of  ani/  tiro 
loci,  a  and  fi  representing  any  functions  of  x  and  y,  and  k  he  any  arbitrary 
constant,  a  +  kfi  =  0 

passes  through  all  their  points  of  intersection. 

So  long  as  Ic  is  arbitrary,  (3)  will  represent  any  straight  line 
through  the  intersection  of  (1)  and  (2),  and  as  h  may  have 
any  value,  it  may  be  determined  so  that  (3)  shall  fulfil  any 
reasonable  condition.     Thus : 


54  ANALYTIC   GEOMETRY. 

First.  To  find  the  equation  of  a  straight  line  passing  tliroiigh 
the  intersection  of  two  given  straight  lines  and  also  through  a 
given  point.  Let  (1)  and  (2)  be  the  two  given  lines  and  {x',y') 
the  given  point.  Then  (3)  is  a  straight  line  through  their  inter- 
section. Since  this  line  is  to  pass  through  (x',y'),  we  have 
A.v'  +  By'-}-C-\-k  {A'x'-{-B'y'  +  C')  =  0,  in  which  everytliiug 
is  known  hut  A;.  Determining  Jc  from  this  equation  and  substi- 
tuting its  A'alue  in  (3),  we  have  the  required  equation. 

Second.  To  find  the  equation  of  a  straight  line  passing  through 
the  intersection  of  two  given  straight  lines,  and  parallel  (or  per- 
pendictdar)  to  a  given  line.  Let  (1)  and  (2)  be  the  given  lines 
and  y  =  mx  +  b  the  line  to  which  (3)  is  to  be  parallel  (or  per- 
pendicular).     Solve   (3)  for  y  and    place  the  coefficient  of  x 

equal  to  m  [or ] ;  from  this  equation  determine  Ji  and  si;b- 

stitute  its  value  in  (3).     The  resulting  equation  will  be  the  line 
required. 

Examples.     Write  the  equations  of  the  following  lines  : 

1.  Through  the  intersection  of 

.T  +  2?/  -  5  =  0  aud  y  -  3.i-  -f  H  =  0, 
and  the  point  (6,  4). 

Substituting  .r  =  G,  //  =  4,  in  x  +  2  y  —  5  +  k  (;/  —  o  .r  +  8)  =  0,  we  find 
^  =  f .     Hence  the  required  line  is  ?/  —  x  +  2  =  0. 

2.  Through  the  intersection  of 

2 a;  +  ?/—  7  =  0  and  x  +2y  —  o  =  0, 
parallel  to  G  x  —  3  y  +  5  =  0. 

We  have  2x  +  ;/  —  1  +  k  (.r  +  2 y  —  5)  =  0.     Solving  for  //, 

^  1  +  2  /.•  "^      1  +  2  A.-" 

Solving  the  parallel  for  y,  ?/  =  2t+ J.     Hence  — - — -—=2.   .-.  A-  =  — <, 
and  the  required  line  is  2  x  —  y  —  [>■-=  0.  "*" 

3.  Tlirough  the  intersection  of 

2x  +  ?/  -  7  =  0  nnd  y  —  x  —  l  =  0, 
perpendicular  to  Sx  +  3//  —  1  =  0.  Aus.    y  —  x—  I  =  0. 


THE   RECTILINEAR    SYSTEM. 


55 


4.  Through  the  intersection  of 

y  —  x  —  l  =  0  and  ?/  —  2  a;  +  1  =  0, 
parallel  to  ?/  =  4  x  +  7.  Ans.    y  =  4:X  —  5. 

5.  Through  the  iuterseetion  of 

y  =  Sx  -\-  14  and  y  =  x-}-(}, 
making  an  angle  of  45°  with  y  =  2x.  Ayis.    y  =  —  Sx—  10. 

6.  The  line  y=mx  +  h  passes  through  the   intersection  of 
y  =  m'x  +  jy  with  ?/  =  vi"x  +  h".     Find  the  value  of  m. 


DISTANCES    BETWEEN    POINTS     AND    LINES,    AND 
ANGLE-BISECTORS. 

38.  To  find  the  distance  of  a  given  X)oint  from  a  given  straight 
line. 

Let  a- cos  a  + ?/ sin  a=p  be  the  given  line  and  (x',y')  the 
given  point.  Through  the  given  point,  P',  draw  ST  parallel 
to  the  given  line  MN.  The  perpendiculars  OQ,  OR,  from  the 
origin  on  these  lines,  coincide  ;  therefore  a  is  the  same  for  both 
(Art.  26),  and  the  equations  of  the  parallels  will  differ  only 
in  the  lengths  of  the  perpendicnlars.  Hence,  if  OR^=p\  the 
equation  of  ST  will  be 

X  cos  a  +  y  sin  a  - 

and  since  P'  is  on  ST, 

x'  cos  a  4-  ?/'  sin  a 


P\ 


\^^ 


P- 

Now  DP'  =  QR  is  the  difference  be- 
tween 2^'  ^ud  2'>i  hence  the  required  dis- 
tance is 

Z)  =  x'  cos  a  +  y'  sin  a.  —p. 

lint  this  is  simply  what  the  equation 
of   the   given  line   becomes  when  j^   is 
transposed  to  the  first  member  and  x\  y',  substituted  for  x,  y. 
Hence,  to. find  the  distance  of  a  given  point  from  a  given  line, 


Fig.  34. 


(1) 


56  ANALYTIC    GEOMETllY. 

put  the  equation  of  the  given  line  under  the  normal  form,  trans- 
pose the  absolute  term  to  the  first  member,  and  siibstitute  the  co- 
ordinates of  the  given  i^oint.  Since  to  put  Ax  +  By  +  C  =  0 
under  the  normal  form  we  divide  by  ^A-  +  J3-,  we  have 

jy  ^  Ax'  +  By'  4-  C 

As  only  the  distance  DP'  is  required,  it  is  not  necessary  to 
attend  to  the  sign  of  ^A--\-B-.     If,  however,  we  follow  the 

general  rule  of  signs  for  putting  the  general  under  the  normal 

n 
form  (Art.  27),  the  last  term  of  (1),  —  .  ivill  always  he 

VA  +  B' 

negative,  since,  when  transposed,  it  must  equal  -\-p.      Now  if 

we  make  x'  and  y'  zero,  (1)  will  be  the  distance  of  the  origin 

C 
from  the  line  =  — 1:^:2=^= ;    hence  the  origin   is   always  con- 

VA  +  B'  °  ^ 

sidered  as  being  on  the  negative  side  of  the  line.     Whenever, 

then,  for  any  given  point,  (1)  is  negative,  the  point  is  on  the 

same  side  of  the  line  as  the  origin.     Thus,  suppose  the  equation 

of  MN  is  2.T  -f-  ?/  —  2  =0.     Dividing  by  Vo,  the  normal  form  is 

2x         V  2 

— ^  +  -^ r.  =0.     Substituting  the  coordinates  of  P',  (|,  f), 

Vo      Vo      Vo 

9     3.  4-  -3.  _  9  '}]_ 

D  =  ^  '  2  ^2 1  _  -'3  _  i)p\ 

Vo  v'o 

This  being  positive,  P'  is  on  the  opposite  side  of  tlie  line  from 
the  origin.     But,  substituting  the  coordinates  of  P",  (— f,  2), 

9      3.    I    o 9  Q 

D  =         '  -  =  —        =  P "  D' 

V5  Vo 

This  being  negative,  P"  is  on  the  same  side  of  the  line  as  the 
origin. 

Were  the  axes  oblique,  tlie  equation  of  the  given  line  being  a;co8a  +  y  cob  j3— p  =  0 
(Art.  26),  as  the  reasoning  above  is  independent  of  p,  x'cos  a  +  j/'cosp  —p  would  be  the 
required  distance. 

39.  The  distance  from  a  given  point  to  a  given  line  may  also 
be  found  as  follows  :  Write  the  equation  of  a  line  through  the 


THE   RECTILINEAR    SYSTEM.  57 

given  point  perpendicular  to  the  given  line  ;  find  the  intersection 
of  the  perpendicular  and  the  given  line  ;  then  find  the  distance 
from   the   given    point    to    this    intersection   by   the    formula 

d  =  V(a;'  -  x"y-{-  (y'  -  y"y.  Thus,  to  find  the  distance  from 
(8,  1)  to  3. r  — 42/4-5  =  0;  the  perpendicular  through  (8,  1) 
to  dx  —  ■iy-{-o  =  0  is  y  —1  =  —^{x—8);  combining  this  with 
dx  —  iy-{-o  =  0,  we  have  for  the  point  of  intersection  (5,  6). 
Hence  d  =  V(8-5)2+ (1-5)2=5.  This  method  is  usually 
less  expeditious  than  that  of  Art.  38. 

Examples.  Determine  the  length  of  the  perpendicular  from 
the  point  to  the  line  in  the  following  cases,  ascertaining  in  each 
case  whether  the  point  and  the  origin  are  on  the  same  or 
opposite  sides  of  the  line. 

1.  3a; 4- -1^-2  =  0,  (2,  7). 

Alls.  -V2- ;    on  the  opposite  side  from  the  origin. 

2.  3.^•  — 42/4-5  =  0,  (8,  1). 

Ans.  5  ;  on  the  side  of  the  origin. 

3.  4ic-3^-G  =  0,  (1,  -1). 

Ans.  i ;  on  the  opposite  side  from  the  origin. 

4.  Sx  +  iy  +  2  =  0,  (2,4). 

Ans.  ^  ;  on  the  side  of  the  origin. 

5.  y—'2x+ 1=0,  (-1,  -3).  Ans.  0. 

6.  Find  the  lengths  of  the  altitudes  of  the  triangle  whose  sides 
are  Ax-Sy  +  8  =  0,  12.r- 5?/ 4-8  =  0,  2x~y  =  0. 

Ans.  The  vertices  are   (1,  4),   (  —  4,  —8),   (4,  8),  and   the 

,..,    ,        2        16     16 

altitudes  ,    — ,    — 

V5      5'    13 

7.  Find   the  length  of   the    altitudes  of   the  triangle  whose 

vertices  are  (1,2),   (-2,0),   (6,  -1). 

.          19         19         19 
Ans.   ■ — r=,     — :=? 


Vl3     V65     V34 


58 


ANALYTIC    GEOMETEY. 


8.  Find  the  area  of  the  triangle  whose  vertices  are  (2,  3), 
(-1,4),   (6,5). 

The  line  througli  (—1,  4)  and  (G,  o)  is  :•■  —  7  //  +  20  =  0  ;  its  normal  forni. 

r  7  ;/  '^n  1  n 

IS  — '-  +  -!^ -^  =  0.     The  distance  of  (2,  3)  from  this  side  is  =-^. 

V50      Vso      VSU  v^ 

The  length  of   the  line  joining  (—  1,  4)  with  (0,  5)  is  VdO.     Hence  the 

area  =  ).  (  -^  X  V50  )  =  5. 
'  VV5U  / 

9.  Find  the  area  of  the  triangle  whose  sides  are  2.r  +  ?/—  7  =  0, 
y-x-\=0,  x  +  2y-o  =  0. 

Ans.  The  vertices  are  (3,  1),  (2,  3),  (1,  2),  and  area  f. 

10.  Find    the   distance    between    the    parallels   y  =2x  —  i), 

y  =  2x  +  8. 

The  line  3/ =  2  a-  — 6  crosses  Y  at  (0,  —0);  the  distance  of  this  point 
14 


from  ?/  =  2  .r  +  8  is  — 


^/5 


11.  Find  the  distance  between  the  parallels 
y=3 X,  y  =  ox—10. 


Ans.   VlO. 


"40.    To  Jjnd  the  equation  of  a  line  bisecting  the  angle  beticeen 
tico  given  lines. 

Let  a;  cos  a  +?/sinu  —  ^^  =0,  (1) 

xcosa' -\-y  sina' —]>' =  <\  (2) 

be  the  two  given  lines.     Then 

{x  cos  a'  +  y  sin  a'  — p')  -f-  h:  {x  cos  a  +  2/  sin  a  — p)  =  0      (3) 

is  a  straight  line  throngh  their  intersection.     Now  the  quantities 

in  the  parentheses  are  the  distances 
of  anv  point  (.r,  y)  from  the  lines  (2) 
and  (1)  (Art.  38).  Thus,  if  J/JV, 
31' y,  be  the  lines  given  bj'  (1)  and 
(2).  then  (3)  is  the  equation  of  so?»e 
line  VP  through  their  intersection  V, 
nnd  the  parentheses  are  the  distances 
i'D\  PD,  of  any  of  its  points  from 
Pigg.  M'N'  and  3IN.      Now  if  A"  =  -1, 


THE    1IECTIL[NEAR    SYSTEM.  59 

PD' =  PD  from  (3),  and  (3)  will  be  the  equation  of  the  line 
bisecting  the  angle  MVN'.  AVhen  a  point  P  is  on  the 
same  side  of  a  line  as  the  origin,  we  have  seen  that  the 
perpendicular  PD  is  negative  (Art.  38).  For  the  angle 
MVN\  P  is  on  the  same  side  of  both  lines  that  the  origin  is, 
and  hence  both  perpendiculars  must  be  negative,  that  is,  have 
the  same  sign,  or  A:=  — 1.  For  the  angle  N'VN^  Q  is  on 
the  same  side  of  one  line  that  the  origin  is,  but  on  the  opposite 
side  from  the  origin  in  the  case  of  the  other  line  ;  one  perpen- 
dicular must  therefore  be  negative,  and  the  other  positive,  that 
is,  have  opposite  signs,  or  ^'=1.  Hence,  to  bisect  the  angle 
betiveen  tivo  given  lines,  put  their  equations  under  the  normal 
form,  and  subtract  or  add  them  according  as  the  origin  does  or 
does  not  lie  ivithin  the  angle  to  be  bisected. 

Examples.  1.  Find  the  bisector  of  the  angle  between 
12  a;  +  5  ?/ —  2  =  0  and  3  .^' —  4  y/ + ''^  =  ^1  in  which  the  origin 
lies.  Ans.  %^ri6^71y — 4W=0. 

2.  Find  the  bisectors  of  the  angles  between  2  a;  -^'y  +  8  =  0, 
x  +  2y  — 3  =  0.  Ans.     3.T  +  3  ?/  +  5  =  0;  cc  — y+11  =0. 

3.  Find  the  bisectors  of  the  angles  between  2.^•4-2/ +  8  =  0, 
and?y  =  0.  Ans.   2  ;r +  (1  ±  V5)//+ 8  =  0. 

4.  "Write  the  equations  of  the  bisectors  of  the  angles  between 
the  axes  y  =  0,  x  =  0.  Ans.  y  ±  x  =  0. 

5.  Of  what  line  would  Eq.  (3),  Art.  40,  be  the  equation  if 
k  =  2?  if  A-  =  n ?  ^  / 

'       6  - 


60 


ANALYTIC    GEOMETRY. 


SECTION  v.— THE  POLAR  SYSTEM. 


41.  Derivation  of  polar  from  rectangular  equations.  When 
the  pole  is  taken  at  the  origin  and  the  polar  axis  is  coincident 
with  the  axis  of  X,  auy  rectangular  equation  of  a  straight  line 
may  be  transformed  into  the  corresponding  polar  equation  (that 
is,  the  polar  equation  expressed  in  terras  of  the  same  constants) 
by  means  of  the  relations  a;  =  r  cos  ^,  y  =  ?•  sin  ^  (Art.  23). 
The  simplest  and  most  useful  of  the  polar  equations  is  the 
normal  form. 


42.  Normal  polar  equation  of  the  straight  line.  The  normal 
rectangular  form  being  x  cos  a+^sin  a=p,  substituting  a;=rcos  6 
and  y  =  r  sin 6,  we  have  r(cos 0  cosa  +  sin 0  sina)  =p,  whence 


r  = 


P 


Discussion.     If  6=0°,  r  = 


cos  {0  —  a) 
P 


P 


cos  (  —  a)         cos  a 

the   point    Q  where  MN  crosses  the   polar   axf| 


(1) 

OQ,  locating 
If  0  =  a, 


r  = 


P 


cosO°  ~-^^'  g^^'i^g  the  point  D.     If  $>a  and  increasing, 

6  — a  is  increasing.  cos(^  — a)  decreases,  and  hence  r  increases 
till  6*  =  a  +  00°,  when  cos(^  -  a)  =  cos  90°  =  0  and  r  =  x ,  as  it 

should  be,  since  r  is  then  parallel 
to  3IN  and  must  be  produced 
infinitely  to  meet  the  line.  When 
$>a  +  00°,  ^-a>yO°,  and  r  is 
negative,  showing  that  it  must  be 
produced  backwards,  or  away  from 
the  end  of  the  measuring  arc,  to 
inect  MN,  and  remains  negative 


\ 


\ 


'V^" 


Fig.  36. 


THE   POLAR    SYSTEM.  61 

till  0  =  a-{-  270°,  or  6  —  a  =  270°,  when  r  =  co  agaiu,  and  is  par- 
allel to  J/iV^.     For  ^  =  360°,   r  = j^ — -  =  ^^=0Q.     The 

cos  (—a)         COS  a 

entire  line  is  traced  for  values  of  6  between  0°  and  180°,  for 

which  latter  value  of  0,  r  = ,,  ,,,,o :  = =  OQ. 

cos  (180   —u)  cos  a 

If  MN  is  perpendicular  to  the  polar  axis  and  lies  on  the  right 

P 

of  the  pole,  a  =  0°,  and  the  equation  becomes  r  = -^  ;  if  on 

^  ^  cos  6 

the  left  of  the  pole,  a  =  180°,  and  the  equation  becomes 

p  2^  ~P 


r  = 


cos  (^-180°)       cos  -  (180°  -  6^)       cos^ 


p 


P 

Hence  r  =  ± ;;  is  the  equation  of  all  perpendiculars  to  the 

cos  6  ^         M    A 

polar  axis,  the  negative  sign  applying  to  those  which  lie  on  the 
left  of  the  pole. 

If  the  line  is  parallel  to  the  polar  axis  and  above  it,  a  —  90 
and  the  equation  becomes 

P  P  P    . 

''  ^  cos  (6  -  yO°)  ^  cos -(90°-^)  ^  sln^ ' 

if  below  the  polar  axis,  a  =  270°,  and 

P  P  P 


r  = 


cos  (^-270°)       cos-(270°-^)  sin  ^ 


Hence  r  =  ±  —. — -  is  the  equation  of  all  parallels  to  the  polar  axis, 
the  negative  sign  applying  to  those  which  lie  below  the  pole. 

If  the  line  passes  through  the  p)ole,  p)  ~  0,  and  r  =  0,  except 
when  6  =  90°  +  a,  in  which  case  r  =  -  ;  that  is,  r  is  zero  for  all 

values  of  6  except  when  the  radius  vector  coincides  with  the  line, 
when  r  may  evidently  have  any  value. 

Examples.     1.    Write  the  polar  equation  of  a  line  whose  dis- 
tance from  the  pole  is  5,  the  perpendicular  being  inclined  45"  to 


62  ANALYTIC    GEOMETRY. 

the  polar  axis.     Find  the  intercept  on  the  axis,  and  the  values 
of  0  for  which  r  is  infinite. 

^"^-  '■= ra — T^A  '  5\/2;  135°;  315°. 

cos  {6  —  45  )  '  ' 

2.    Write  the  polar  equations  of  lines  for  which  p  =  2,  a=  60°  ; 
p  =  10,  a  =  120°  ;  and  find  their  intercepts. 

2  8 


3.    Construct  r  — 


cos(e-30°)  cos  (^-(30°) 


4  5 

4.    Construct  r  =  ± ;  r  =  ± 


cos  0  sin  6^ 

5.    Write  the  polar  equations  of  the  sides  of  a  square  whose 

centre  is  at  the  pole  and  side  10,  one  side  being  parallel  to  the 

axis. 

3  —  5 

G.    Find  the  rectangular  equations  of  r  =  — ^ — ;  r 


7.    Find  the  rectangular  equation  of  r  = 


cos  6  sin  0 

9 


cos  {0  —  45°) 
Ans.  x-\-y  —  9V'2  =  0. 

8.  Find  the  polar  equation  ofSa;  —  4?/4-l  =  0. 

Qx 4  V  4-  1 

If  the  normal  form  is  required,  -^^—^ —  =  0,  whence  p  =  h  aii*l 

—  5 

a  —  cos"*  -|,  which  may  be  found  from  the  tables.     Then  substitute  p  and  a 

in  r  = f- — ■ —    If  the  normal  form  is  not  specified,  substituting  directly 

cos(0  — a)  , 

the  values  of  :c  —  r  cos  6  and  y  =  r  sin  6,  we  have  r  = 

4  sin  0  —  3  cos  d 

9.  Find  the  polar  equation  of  y  =  3  x-^'2. 

Ans.  r- 


sin  ^  —  3  cos  B 


<^"l-  ^^  ' 


APPLICATIONS.  63 


SECTION   VI.— APPLICATIONS. 


43.  Recapitulation.  The  foregoing  formulfe  and  equations 
relating  to  points  and  straight  lines  constitute  the  elementary 
tools,  as  it  were,  of  analytic  research  on  the  properties  of  recti- 
linear figures.  The  student  must  remember  that  it  is  not  the 
object  of  Analytic  Geometry  to  produce  these  equations  and 
formulas,  but  to  investigate  the  properties  of  loci  by  means  of 
them.  While,  therefore,  familiarity  with  these  expressions  is 
indispensable,  a  mastery  of  analytic  geometry  implies  a  knowl- 
edge of  their  use  in  the  discovery  of  geometrical  truths  ;  that  is, 
the  mastery  of  a  method  of  research.  The  more  important  of 
these  expressions  are  here  collected  as  a  review  exercise.  The 
student   should   memorize  them,   and   be  able  to  explain  the 

x'  4-  x" 
meaning  of  all    the    quantities    involved.      Thus,    x  =  — — — , 

v'+v"  ^ 

y  =  - — 7-=^,  are  the  equations  of  a  point  midway  between  two 

given  points,  in  which  x,  y,  are  the  coordinates  of  the  required 
middle  point,  and  x',  t/',  a;",  y",  the  coordinates  of  the  given 
points. 

aj  =  ^-±^,    y  =  t±JL.  Equation  (3),  Art.    6. 


f\2 


d  =  Vr'-+  r"--  2  r'r"  cos  {$"  -  0') .  ' ' 

x  =  Xa-{-Xi,  y  =  y^,  +  yi.  " 

a;=>-cos^,  y  =  rsmO^  " 

Ax-\-By+C=0.  " 

O        0 

y=imx  +  h.  " 


(1),  - 

7. 

(1)^  " 

13. 

(2),     " 

22. 

(4),     " 

23, 

(1),     " 

25, 

(1),     " 

26. 

(2),     " 

26. 

64  ANALYTIC   GEOMETRY. 

X  cosa  +  y  sin  a  =2>-  Equation  (3) ,  Art.  26. 

y  —  y'  =  vi{x  —  x') .  " 

y-y'  =  ^~^„{^-^')-  " 


X'  —  X 

c  = 


m'  —  m 


1  +  mm' 


(( 


y-y  = ,         (x  -  x').  « 

1  —  mc 

,  1 

m  =  m  ,    m  = • 

m' 
y  —  y'  =:  m  (x  —  x') .  *' 

y-y'  =  —  {x  —  x').  " 

m 
^,^Ax'+By'-\-C_ 

■Va'+  B' 

acosa'  +  y  sina'— p'±  (ajcosa  +  y  sina— p)  =  0. 

r= ^ "         (1), 

cos(^-a)  ^  ' 


(2>, 

'•    31 

(4), 

"    32 

(1). 

"    33 

(2), 

"    34 

"    35 

(1), 

"    35 

(2), 

"    35 

(1). 


38. 

40. 
42. 


PROPERTIES    OF   RECTILINEAR   FIGURES. 

44.  1.  The  diagonals  of  a  square  are  j^erpendicular  to  each 
other. 

Take  two  adjacent  sides  for  the  axes.  Then,  if  a=  side,  the 
vertices  are  (0,  0),  (a,  0),  (a,  a),   (0,  a),  and  the  equations 

of  the  diagonals  are  y  =  x,    y  =  —  x  +  a,  in  which  m  = ^ 

(Art.  35).  '"^ 

2.  The  line  joining  the  middle  points  of  two  sides  of  a  triangle 
is  parallel  to  the  third  side. 

Take  the  third  side  for  the  axis  of  X,  and  the  origin  at  its 
left-hand  extremity.  Tlien  (0,  0),  (o,  0),  (&,  c)  are  the  ver- 
tices, [-,  Hi  ["  ,  -),  the  middle  points,  and  2/=|  is  the 
line  joining  them. 


APPLICATIONS. 


65 


3.    Tlie  diagonals  of  a  parallelog)xim  bisect  each  other 

With  the  axes  as  in  the  figure,  let  the 
side  OB  =  a,  the  altitudi'  mD=b,  and 
Om  =  c.  Then  the  coordinates  of  C  are 
(a  4-  c,  &) ,  and  the  middle  point  of  OC  is 

'^       The   coordinates   of   B  are 

(a,  0),   of  D,   (c,  h),  and  of  the  middle 

''a  +  c     6' 

2     '    2. 


Fii?.  37. 


point  of  BB, 


4.    T/te  straight  lines  joining  the  middle  points  of  the  opposite 
sides  of  any  quadrilateral  bisect  each  other. 

Let  CO,  0),  (a,  0),  (b,  c),  {d,  e)  be  the  vertices  0,  -B,  C,  D, 
in  order.     Then  the  middle  point  of  each  line  is 

''a-\-b  -{-d     c-\-  e"^ 


5.  Prove  that  the  middle  point  of  the  line  joining  the  middle 
points  of  the  diagonals  of  any  quadrilateral  is  the  point  of  inter- 
section of  the  lines  of  Ex.  4. 

6.  The  lines  joining  the  middle  points  of  the  adjacent  sides 
of  a  parcdlelogram  form  a  parallelogram. 

With  the  notation  of  Ex.  3,  the  slope  of  the  lines  joining  the 

middle  points  of  DC  and  BC,  DO  and  OB,  is  ;  hence 

these  lines  are  parallel. 

7.  The  middle  point  of  the  hypothenuse  of  any  right-angled 
triangle  is  equally  distant  from  the  vertices. 

Take  the  axes  coincident  with  the  sides. 

8.  Prove  that  if  A,  B,  C,  be  squares  on  the  sides  of  a  right- 
angled  triangle  OJiQ,  and  OT  is  perpendicular  to  RQ,  then  RS, 
QP,  and  OT  meet  in  a  point.  With  the  axes  as  in  the  figure, 
let  c  =  OQ,  d  =  OB,  the  sides.  Then  the  coordinates  of  S 
and  B  are  (c,    —  c),   {— d,  0),.  and  the   equation  of  SB  is 


^./ 


66 


ANALYTIC   GEOMETRY. 


X  — 


cd 


4-  d        c  +  d 


.Similarly     the     equation     of     QP    is 


y  = ^^  x  —  c.     The  equation  of  RQ  is  ~ — \-  -^  = 

d  J  —d      —  c 


1,  and 


of  OT,  perpendicular  to  it,  y  =  -x.     Substituting  this  value  of 

y  in  the  equations  of  BS  and  QP,  the  values  of  x  are  found  to 
be  the  same  ;  hence  OT  intersects  them  both  at  the  same  point. 


Fig.  38. 


9.    The  altitudes  of  a  triangle  meet  in  a  point. 

Take  the  axes  as  in  the  figure,  and  let  AB  =  c,  C  being  given 
as  (x',  y') .     Then  the  altitude  through  C  is 


X  =  X' 


(1) 


r_  y 


The  equation  of  BC  is  y  —  y'=  —^ —  (x  —  x') ,  and  that  of  the 
altitude  through  A  is 


Z/  = 


c  —  X' 


-X. 


y 


(2) 


The  equation  of  AC  is  ?/  =  ^a;,  and  of  the  altitude  through 

B   13 

y  =  -^(x-c).  (3) 

y 

Combining  (2)  and  (3)  to  find  their  intersection,  we  obtain 
x  =  x\  which  satisfies  (1).  Hence  (1),  (2),  and  (3)  meet  in 
a  point. 


APPLICATIONS.  67 

10.     The  perpendiculars  erected  at  the  middle  points  of  the 

sides  of  a  triamjle  meet  in  a  point. 

„< 
The  equation  of  AC  (Fig.  30)  is  7/  =  -^  a;,  and  that  of  the 

perpendicular  to  AC  through  B'  is 

y'         x'  /        x'\  , . . 


The  equation  of  BC  is  y  —  y'  =  —^ —  (a;  —  x') ,  and  that  of 

X  —  c 


x' 
the  perpendicular  to  BC  through  A'  is 


y'      c  —  x'f        c  +  x'\  .r,\ 


2  y 

The  perpendicular  to  AB  at  C  is 

X  =  -.  (3) 

2  ^  ^ 

Combining  (1)  and  (2),  eliminating  y,  we  have  .'k  =  —    Hence 
(1)  and  (2)  intersect  on  (3). 

11.    The  medials  of  a,  triangle  meet  in  a  point. 
The  middle  points  A',  B',  C  (Fig.  39),  are 

^c  +  x'  y'\     fx'  y'\     (c 


2  '2;'  w^r  V2'^'' 


and  the  equations  of  the  medials  are 


y  =  -y^x,  AA',  (1) 

c  +  X 


_    y' 


y  =  -r^(x-c),  BB\  (2) 

X  —  2c 
y^y\2x-c)^  CC".  (3) 


2a;'- 


1 


Combining  (2)  and  (3),  we  find  they  intersect  in  f — ^^—,  - 

\     3        3 

and  these  values  satisfy  (1)  ;  hence  (1),  (2),  and  (3)  meet  in 

a  point. 


68  ANALYTIC   GEOMETRY. 

12.    To  find  the  general  analytic  condition  that  three  straight 
lines  may  meet  in  a  point. 

Let  (1)  y  =  m'x  +  b',  (2)  y  =  m"x  +  h'",   (3)  y  =  m"'x+h"' 
be  the  three  lines.     The  intersection  of  (1)  and  (2)  is 

jj"-h'  m'b"-b'm" 


X  =  ■ ,      y  ■ 


m'—m"  m'—m" 

But  these  must  satisfy  (3)  ;  hence 

m' b" -  m" b'  +  m"'b' -  m'b'"  +  m" b'" -  m'" b"  =  0. 

13.  Shoiv  that 

11?/  — 20.r-8  =  0,  y-4x  =  0,   13y  -  28.x- —  8  =  0, 
meet  in  a  point. 

14.  To  find  an  expression  for  the  area  of  a  triangle  in  terms 
of  the  coordinates  of  its  vertices. 

Let  {x\y'),  {x",y"),  (.r'",  ?/'"),  be  the  vertices.    The  equation 

of  a  line  through   the   first  two  is  y—y'—^ — ^(x  —  x'),  or 

.^  —  a;' 

(y"—y')x-i-(x'—x")y-\-y'x"—y"x'=0.       Hence  the  perpen- 
dicular distance  from  this  side  to  (x'",  y'")  is 

(y>'_  y<)  x"'+  (x'-x")  y"'+y'x"-  y"x' 
'  ■\/(y'-y"y  +  {x'-x"y         ~' 

But  the  denominator  of  this  expression  is  the  distance  between 
(x',  y')  and  {x",  y").     Hence  the  area 

=  Ibase  X  altitude  =  i  [.t' (?/'"-//") +-'^""(?/'-2/"') +«"'(?/"-y)]- 

15.  Find  the  area  of  the  triangle  tvhose  vertices  are  (2,  3), 
(-1,4),  (6,5). 

16.  Fiiid  the  equation  of  a  straight  line  passing  through  a 
given  point  and  diriding  the  line  joining  tivo  given  points  in 
a  given  ratio. 

Substitute  in  y  —  y'=  •-, — ^,  (x  —  x')  for  .r",  y'\  the  values  of 

x—x' 


APPLICATIONS.  69 

X  and  y  in  Equation  (2),  Art.  6,  and  for  x',  i/',  the  coordinates 
h,  k,  of  the  given  point,  and  we  have 

^  m{x"-h)  +  n{x'-h)^  ' 

17.  The  bisectors  of  the  interior  angles  of  a  triangle  meet  in  a 
point. 

Let  the  equations  of  the  sides  of  the  triangle  be 

iKcosa'  -\- y  aiua'  —j)'  =0,  (1) 

a;  cosa" +// sina"  —  y)"  =  0,  (2) 

a:cosa"'+?/sina"'-iV"=0,  (3) 

and  let  the  origin  be  tvithin  the  triangle.  Then  the  origin  lies 
within  each  of  the  three  angles  to  be  bisected,  and  the  equations 
of  the  bisectors  (Art.  40)  are 

xcosa'  +i|/siua'  — ^/  —  (x  cosn" -\- y  s'ma"  —  j)")  =  0,  (4) 
cccosa"  +  y/sina"  - p"  -  {x  COSa'"  +  y  nina'" - p'")  =0 ,  (5) 
a;cosa"'+?/ sina'"  — //"— (.^  cosa'  -\- y  sina    —  p')   =0.      (6) 

But  values  of  x  and  y  which  satisfy  any  two  of  these  equations 
also  satisfy  the  third  ;  hence  these  three  lines  meet  in  a  point. 

/    18.    The  lines  tvhich  pass  through  the  vertices  of  a  triangle  and 

bisect  the  angles  supplemental  to  those  of  the  triangle  meet  in 

a  point. 

F'or   brevity,   represent   by   a  =  0,    /3  =  0,    y  =  0,   Equations 

(1),    (2),   and   (3)    of  Ex.    17.       Then  a  +  /3  =  0,   (3  +  y^O, 

y  +  a  =  0  are  the  bisectors  required. 
-^ 

19.  The  bisectors  of  any  two  exterior  angles  of  a  triangle  and 
of  the  third  interior  angle  meet  in  a  point. 

The  bisector  of  the  exterior  angle  of  (1)  and  (2),  Ex.  17, 
is  a  +  ^  =  0,  and  of  (2)  and  (3)  is  /?  +  y  =  0,  and  the  bisector 
of  the  interior  angle  of  (1)  and  (3)  is  a  — y  =  0.  Subtracting 
the  second  of  these  equations  from  the  first,  we  have  the  third. 


70  ANALYTIC    GEOMETRY. 

20.  Tlie  bisectors  of  the  angles  betioeen  the  bisectors  of  perpen- 
diculars are  the  perpendiculars  themselves. 

Let  y  =  mx  +  6,  y  = x  +  b',  be  the  perpendiculars.    Their 

normal  forms  are 

y  —  mx  —  &  _  n  my  +  x  —  mb'  _  ^ 

vr+w"-     '       Vi + m' 

and  their  bisectors  are  y  —  mx  —  b±  {my  +  x—mb')  =  0.      The 
normal  forms  of  these  latter  are 

(1  +m)y  +  (l  —m)x—(mb'+b)  _^ 
V(l+m)2  +  (l-m)2 

(1  -  m)  ?/-(!+  m)  X  +  {mb'—  b)  _  q 
^{l+my  +  {l-my 

and  their  IMsectors  are 

(1  +  m)  ?/  +  (1  —  vi)  X  —  {mb'+h) 

±[(1  -  ra)  z/  -  (1  +  m)  x  +  {mb' -  6)]  =  0, 

or  ?/  —  mx  —  &  =  0,  and  my  +  x  —  mb'=  0,  which  are  the  given 
perpendiculars. 

21.  To  find  the  condition  that  the  three  poiyits  {x\  y'), 
{x",  y"),  {x"\  y'"),  shall  be  coUinear,  i.e.,  lie  on  the  same 
straight  line. 

22.  Prove  that  the  line  ivhich  divides  two  sides  of  a  triangle 
proportionally  is  parallel  to  the  third  side. 

^  9,  >  '■ 


CHAPTER  III. 

EQUATION   OF   TEE  SECOND   DEaREE. 
THE   CONIC  SECTIONS. 

o-oXXoo 


SECTION   VII. —COMMON    EQUATIONS    OF   THE 
CONIC   SECTIONS. 


45.  The  Conic  Sections.  It  has  been  shown  in  the  previous 
chapter  that  every  complete  equation  of  \hQ  first  degree  between 
X  and  2/?  Ax  -f-  Bij  -f-  0==  0,  and  the  various  forms  which  such 
an  equation  may  assume  owing  to  a  change  in  tlie  values  or 
signs  of  the  arbitrary  constants  A^  B,  C,  is  the  equation  of  a 
straight  line.  In  the  present  chapter  it  will  be  shown  that 
every  equation  of  the  second  degree  between  x  and  y, 
Ay^ -\- Bxy  +  Cx- -{- Dy -\- Ex  +  F  —  0^  and  the  various  forms 
it  may  assume  when  diilerent  values  and  signs  ai'e  given  to 
the  arbitrary  constants  A,  B,  C,  D,  E,  F,  represents  some 
one  of  a  family  of  loci  called  the  Conic  Sections.  These  loci, 
which  for  brevity  may  be  designated  Conies,  are  so  named 
because  every  section  of  the  surface  of  a  right  cone  with  a 
circular  base  by  a  plane  is  one  of  this  family. 

They  may  all  be  traced  by  a  point  so  moving  that  the  ratio 
of  its  distances  from  a  fixed  point  and  a  fixed  straight  line  re- 
mains constant,  the  particular  locus  traced  depending  upon  the 
value  of  this  constant.  Since  all  the  loci  of  this  family  may 
thus  be  generated  b}'  a  pouit  moving  under  a  single  law,  it  will 
evidently  be  possible  to  express  this  law  in  a  single  equation, 
and  to  derive  the  particular  cases  from  this  general  equation 


72 


ANALYTIC    GEOMETRY. 


by  assigning  the  corresponding  value  to  the  ratio.  The  proof 
of  the  foregoing  statements  and  the  discussion  of  the  general 
equation  is,  however,  greatly  facilitated  by  a  knowledge  of  the 
forms  and  elementary  properties  of  these  loci ;  we  shall  therefore 
first  determine  their  equations  separately  from  some  of  their 
properties  with  a  view  to  the  discovery  of  their  forms,  reserving 
the  discussion  of  the  general  equation  until  the  student  has 
thus  become  familiar  with  the  various  loci  which  it  represents. 


THE    CIRCLE. 

46.  Defs.  The  path  of  a  point  so  moving  that  its  distance 
from  a  fixed  point  remains  constant  is  a  circle.  The  constant 
distance  is  the  radius,  the  fixed  point  the  centre. 

47.  General  equation  of  the  circle. 

Let  (m,  n)  be  the  centre  C\  R  the  radius,  and  P  any  point  of 

the  circle.    From  the  right-angled  triangle 

PCM,  CP-  =  CM-  +  MP\ 

or       {y-ny  +  {x-mY=R\  (1) 

which  is  the  required  equation.  Hence, 
to  tvrite  the  equation  of  any  circle  ichose 
X>osition  and  radius  are  knoiun,  substitute 
the  given  values  of  m,  n,  and  li,  in  the 
above  equation.  Thus,  the  equation  of 
the  circle  whose  radius  is  6  and  centre  is 
(6,  -2),  is  (y  +  2)-+{x-6y=36,  or  /+a;-  +  4.v-12.T+4  =  0. 
By  assigning  different  values  to  m  and  n,  we  may  derive  the 

equation  of  a  circle  in  any  position  from 
the  general  equation  (1).  Two  of  these 
derived  equations  are  of  frequent  use  and 
should  be  memorized.  First:  when  the 
centre  is  at  the  origin,  in  which  case 
m  =  0,  n  =  0,  ;>nd  ( 1 )  becomes 


Fig.  40 


y-  -t-  .r  =  R; 


(-0 


Fiir. -11. 


CO:SDION    EQUATIONS    OF   CONIC    SECTIONS. 


73 


called  the  central  equation  of  the  ch-cle.     Second :    when  the 
origin  is  at  the  left-hand   extremity  of  any  diameter  assumed 
as  the  axis  of  X,  in  which  case  m  =  M, 
v(,  =  0,  and  (1)  becomes 

y^^-IRx-x".  (3) 

Thus,  the  central  equation  of  the  circle 
whose  radius  is  6  is 

/  +  a"  =  36, 
and  when  referred  as  in  Fig.  42, 

y-  =  \2x-x^. 

The  above  forms  may  be  obtained  directly  from  the  correspond- 
ing figures.  The  student  will  observe  that,  by  transposition, 
either  (2)  or  (3)  shows  that  PM^  =  AM.MA',  a  well-known 
property  of  the  circle  from  which  these  equations  might  have 
been  established. 


Fig.  42. 


48.    The  equation  of  every  circle  is  some  form  of  the  equation 
y^^^+By  +  Ex  +  F=0.  (1) 

Expanding  the  general  equation  of  the  circle 

(y-ny-i-{x-my-=R^  (2) 

we  have  y-  +  xr  —  2  ny  —  2  mx  +  vr  -{-n"  —  R-  =  0,  (3 ) 

which  is  of  the  same  form  as  (1).  The  first  two  terms  of  (3) 
are  independent  of  m,  ?;,  and  R,  so  that  no  change  in  the 
position  or  magnitude  of  the  circle  can  affect  these  terms. 
Every  equation  of  a  circle,  therefore,  tvill  contain  the  squares  of 
x  and  y  loith  equal  coefficients  and  like  signs.  The  remaining 
terms  will  vary  with  the  radius  and  position  of  the  circle.  Thus 
E=0,  when  m  =  0,  and  y~ -\-x^-\- Dy  +  F=0,  is  the  equation 
of  all  circles  whose  centres  are  on  Y\  i)=  0,  when  n  =  0,  and 
y^ -{-X? -\- Ex+F  =  ()  applies  to  all  circles  whose  centres  are 
on  X;  if  both  m  and  n  are  zero,  than  ^  =  0,  D=0,  and  we 
have  the  central  form  y-  +  xr  =  R-,  the  centre  being  at  the  origin  ; 
if  the  origin  is  on  the  curve,  then  the  equation  can  have  no 


74  ANALYTIC   GEOMETRY. 

absolute  term,  or  i^  =  0,  and  (1)  becomes  y^  ^  x^  ^  By  +  Ex  =0  ; 
if  B,  E,  and  F  are  all  zero,  we  have  y^  ^  a^  =  0,  which  is  true 
only  for.T^O,  ?/=0,  the  origin,  the  circle  becoming  a  point; 
this  may  be  regarded  as  the  limiting  case  of  the  central  form 
as  the  radius  diminishes  indefinitely.  ^ 

49.   Conversely,  every  equation  of  the  form 

f-^  .^2+  Dy  +  Ex  +  F={),  (1) 

which  is  not  ivij^ossible,  is  the  equation  of  a  circle. 

Adding  to  both  members  of  (1)  the  squares  of  half  the  co- 
efficients of  a;  and  y,  we  obtain 

f  +  By  +  ^  +  x^  +  Ex  +  ^  =  ^ ^^-  F, 
4  4         4         4 

or  (^y  +  ^'Y+(^a;  +  |J=i(i)^'+£'^-4F),  (2) 

which  is  of  the  same  form  as 

{y-ny+{x-my  =  R\  (3) 

and  in  which,  therefore, 

-f  =  'N    -f  =  *'i.    \{D''+E--^F)  =  R\       (4) 

If  D-+  E-  >4:F,  then  li-  is  positive,  E  is  real,  and  the  equa- 
tion represents  a  circle  whose  centre  is  [  —  ~,   —  — Y  and  whose 

, V      2  2  7 

radius  is  J  V/>^'-t-  E'-iF.     If  D'+  E'=-iF,  then  i?-  is  zero, 

and   the  equation   becomes  |  ?/ +— j  -f- Li- _(- _y=o,  which    is 

F  T) 

satisfied  only  for  x  =  -     ,  .?/=-^-,  or  the  circle   becomes  a 

point,  namely,  the  centre,  which  may  be  regarded  as  a  circle 
whose  radius  is  zero.  If  D-+E''<4  F,  then  E-  is  negative,  E 
is  imaginary,  and  the  equation  is  impossible  since  the  sum  of 
two  squares  cannot  be  negative.  We  have  thus  three  cases,  in 
which   A'   is   real,  zero,  or   imaginary,    and    for    brevity   and 


COMMON   EQUATIONS    OF   CONIC    SECTIONS.  ib 

uniformity  of  expression  we  may  say  that  evenj  equation  of  the 
form  y--{-x"-\-Di/  +Ex+F=0  is  the  equation  of  a  circle,  real 
or  imatjinary . 

The  equation  ay'-\- ax--\-dy +ex  ■\-f=Vi  may  be  reduced  to 
the  form  of  (1),  and  is  therefore  the  most  general  form  which 
the  equation  of  a  circle  can  assume. 

50.  To  determine  the  centre  and  radius  of  a  circle  whose 
equation  is  given. 

When  the  equation  is  given  in  the  form  (^  — «)"+  (x  —  m)"=E-, 
the  centre  {m,  n)  and  radius  B  may,  of  course,  be  determined 
by  inspection.  If  given  in  the  form  y^-{-  a;--f-  Dy  +  Ex  +  F=0, 
we  may  put  it  under  the  above  form  by  adding  to  both  members 
the  squares  of  half  the  coefficients  of  x  and  ?/,  as  in  Art.  49. 
Otherwise,  by  equations  (4),  Art.  49,  the  coordinates  of  the 
centre  are  half  the  coefficients  ofx  and  y  loith  their  signs  changed, 
and  the  radius  R  =  .V  ^^U- + £'-  —  4  F.     Thus ,  given 


y''-^x'-Ay+2x  +  l  =  0,  m  =  -l,  ?i  =  2,  i?  =Wl6+4-4  =  2. 

51.    The   equations  of  concentric  circles  differ  only   in   their 
absolute  terms. 

Since    the   values   of   m(  =  — —  j  and  ?if  =  — --j  are    inde- 


pendent of  F,  and  i?  =  i  V-D-+-E-— 4i^,  if  in  the  equation 
of  any  circle  F  changes,  D  and  E  remaining  the  same,  the 
circle  retains  its  position  but  changes  its  size.  Hence  circles 
are  concentric  ivhose  equations  differ  only  in  their  absolute  terms. 

Examples.     1.  Write  the  equation  of  the  circle  whose  radius 
is  7  and  centre  at  (0,  8). 

Ans.    (7/-8)-+(.^•-0)^=49,  ov  y-+ x^-\i:,y +lb  =  Q. 

2.  Write  the  equations  of  the  following  circles: 

Centre  at  (  —  1,  —4),  radius  2; 
Centre  at  (0,  0),  radius  9; 

Centre  at  (5,  0),  radius  5; 

Centre  at  (  — o,  5),       radius  5. 


76  ANALYTIC    GEOMETRY. 

3.  Write  the  equation  of  a  circle  whose  centre  is  (6,  8), 
passing  through  the  origin. 

4.  Which  of  the  following  equations  are  those  of  circles? 
x'+  2f-+  8.7; -4^  +7  =  0  ;   .«2+  y-  +  xy  +  x  +  //  -1  =  0  ; 
a-^- 2/-+ X- -22/ +  4  =  0;        /+ ,;--4y- 8.« +  1=0  ; 
x'^^'if+x-Sy=();  2.r+2/+4.r-3^+7  =  0; 
a-2  +  2?/-4.r-l  =  0;  f  ?/-+ f  x2-2a;  =0. 

5.  Write  the  equations  of  a  circle  whose  radius  is  6  when 
(1)  both  axes  are  tangent  to  the  circle;  (2)  when  X' X  is  a 
tangent  and  y  y  passes  through  the  centre  (two  eases). 

6.  Determine  the  position  and  radius  of  the  following  circles  : 

?/2+ar  — 82/  +  4.T— 5  =  0.  Ans.  (  —  2,  4),  5. 

2/2_|_.^^_|.107/-4a;-7  =  0.  Ans.  (2,  -5),  6. 

y'+x--\-\Qy+Ax-'20  =  (),  Ans.  (-2,  -5),  7. 

^2^.^^_2?/+G.^•=0.  Ans.  (-3,  1),  VlO. 

y'i^x'+Zy--x-^^=0.  Ans.  (|,  -|),  4. 

/+a;2+42/-2.T  +  r>  =  0.  Ayis.  (1,  -2),  0. 

362/-+36rr-24?/-36.t;-i;31  =  0.    Ans.  (.^,  i),2. 

?/+  3'-'+  .'/  + .«  - 1  =  0.  Ans.  ( -  1-,  -1) ,  I  V(3. 

/+  a^+  7/  +  .i;  + 1  =  0.  ^ws.  ( -i,  -i) ,  i  V  ^. 

y-+x--  2/  -  .i-  +  4  =  0.  .4»s.  ( 1. ,  I) ,  ^  V^a4. 

7.  Write  the  equation  of  the  circle  whose  centre  is  at  the 
origin,  and  which  touches  the  line  3a;— 4_y +25  =  0. 

Putting  the  equation  of  the  line  under  the  normal  form, 

3a:  ,  4y       . 
o         0 
which  must  equal  /.'.     Tloncc  y-+a-2=25. 

8.  Write  the  cfiuation  of  the  circle  whose  centre  is  (0,  0), 
and  which  touches  the  line  3.a-  +  //— 0  =  0. 

9.  Wri1(>  the  ('((natiou  of  tlu'  circle  whose  centre  is  (2,  3), 
and  wiiicli  tunchos  3.t;  +  4  v +  12  =  0. 


COMMON   EQUATIONS    OF    CONIC    SECTIONS. 


77 


10.  Prove  that  the  sum  of  the  equations  of  auy  number  of 
circles  is  the  equation  of  a  circle. 

11.  Prove  that  if  the  equation  of  a  straight  line  be  added  to 
the  equation  of  a  circle,  the  sum  is  the  equation  of  a  circle. 

52.   Polar  equation  of  the  circle.  /-v  -- 

The  general  equation  of  the  circle  being  {^y  —  nf-\-{x—mf=^R-, 
let  the  pole  be  taken  at  the  origin  and  the  polar  axis  coincident 
with  X.  Then,  if  ?•',  B'  (Fig.  43),  are  the  polar  coordinates  of 
the  centre  C,  and  ?•,  ^,  those  of  any  point  P,  from  the  formulaa 
for  transformation,  Eq.  (4),  Art.  23,  we  have  .'C  =  rcos^, 
y=-r  sin  ^,  m  =  r'  cos  6',  n  =  r'  sin  $',  which  being  substituted 
in  the  above  equation,  there  results,  after  reduction, 


r-2n-'cos(^-^')=^'-'' 


..12 


(1) 


Fig.  43.  Fig.  44. 

From  this  equation  we  may  derive  that  of  the  circle  in  any 
given  position  by  assigning  the  proper  values  to  r'  and  0'. 
Thus,  if  the  centre  is  at  the  pole,  r'=  0,  and  (1)  becomes 

r=E,  (2) 

which  is  true  for  all  values  of  0.      (See  Ex.  1,  Art.  19.)     If 

the  pole  is  on  the  curve,  and  the  polar  axis  a  diameter,  ^'  =  0, 

r'=B,  and  (1)  becomes 

r=2Rcose,  (3) 

which,  being  true  for  all  positions  of  P  (Fig.  44),  shows  that 
OPA\  or  the  angle  inscribed  in  a  semi-circle,  is  a  right  angle. 
(See  Ex.  2,  Art.  19.) 


78  ANALYTIC    GEOMETRY. 

Discussion  of  Equation  (1).    Solving  the  equation  for  r,  we  have 


r  =  r'  coB(9-e')  ±  -y^Jf^-r'-Bin^ie-d'), 

which  gives  two  values  of  r  for  every  value  of  9,  locating  two  points  P  and  Pj,  so  long 
as  Ji->r'-sin-(9~e'),  or  iJ>  r' 8in(e  — S')-  If  Ii<r'  8in{8  —  t)'),  r  is  imaginary.  If 
E=r'  Bm(8  — 6'),  r  has  but  one  value  ;  in  this  case  the  two  points  P  and  Pj  coincide, 
the  secant  OP  becoming  the  tangent  OP',  or  OP",  and  r=  r'  cos(9  — 0')  =  OP'  =  OP". 
Since  this  relation  is  true  only  when  the  triangles  OCP'  and  OOP"  are  right  triangles, 
we  see  that  the  radius  is  perpendicular  to  the  tangent  at  the  point  of  contact ;  this  also 
appears  from  the  condition  7?=  r'  r\b{9  —  6'),  which  must  he  fulfilled  when  r  has  but  a 
single  value,  this  condition  being  CP'  =  OCelaCOP',  or  CP"  =  OCsinCOP". 

Examples.  1.  Write  the  polar  equation  of  the  circle  whose 
radius  is  10,  the  pole  being  on  the  circle  and  the  polar  axis  a 
diameter.  Discuss  the  equation,  showing  that  the  entire  circle 
is  traced  for  values  of  0  from  0°  to  180°. 

2.  Construct  the  circles  whose  equations  are  r  =  8cos^, 
r  =  —  8  eosO. 

3.  Derive  the  polar  form  r  =  2  R  cos  6  from  the  correspond- 
ing rectangular  form  y'  =  2  Bx  —  x-. 

THE    ELLIPSE. 

53.  Defs.  The  path  of  a  point  so  moving  that  the  sum  nf  its 
distances  from  tivo  fixed  p>oints  is  constant  is  called  an  ellipse. 
The  two  fixed  points  are  called  the  foci,  the  point  midway 
between  them  the  centre,  and  the  lines  joining  any  point  of 
the  ellipse  with  the  foci  the  focal  radii. 

54.  Central  equation  of  the  ellipse. 

Let  F,  F'  be  the  foci,  0  the  centre  and  origin,  the  axis  of  X 
being  coincident  with  FF',  P  any  point  of  the  ellipse,  FF'—  2c, 
and  2a  the  constant  sum.       Then  FP  +  F'P=2a.     But 


F 


FP  =  ^P3P+  ME^  =  V(x  +  c)^+  y\ 
F'P=  -^F^I\P+MP-^  ^(x-cy+y'. 


Hence         Vix  +  c)'+  y'+V{x  -  cy+  ?/=  2a. 
Transposing  the  second  term  to  the  second  member,  and  squaring, 
(X  +  c)2+  /  =  4 a^-  4 a  V {x-cf+f'  +  {x  - cf+  y\ 
or  ex  —  '/,- =  —  a  V(.r  —  c^+y'^ 


COMMON   EQUATIONS    OF    CONIC    SECTIONS. 


79 


Squaring  asrain, 

aV+  (a--  c-)x':=  a\a--  c").  (1) 

Discussion  of  the  Equation.  Since  only  the  squares  of 
the  variables  enter  the  equation,  the  ellipse  is  symmetrical  with 
respect  to  both  axes.  Making  y  =  0,  the  X-intercepts  are  ±  a. 
Take  OA==OA'=a,  then  ^^r=2a=the  constant  sum,  and 
as  the  sum  of  two  sides  of  a  triangle  is  greater  than  the  third 
side,  PF+  PF'=2a>FF'=  '2c,  or  A  and  A'  are  icithout  the 


-X 


Fig.  45. 


foci.  Making  x  =  0,  the  T-intercepts  are  ±  Vo^—  o",  which  are 
real,  since  a  >  c,  and  locate  the  points  B,  B'.  Solving  the 
equation  for  y, 


y  =  ±  -^  {a'-  c'){ct'-  x"), 


a 


which  is  imaginary  if  x  >  a  numerically,  and  therefore  the  curve 
lies  wholly  within  the  limits  A  and  A'  along  X.     Solving  for  x, 


X 


=  *^^^f^ 


r 


r 


which  is  imaginary  if  3 — 72  >  1,  or  y  >  Vfr—  c-  numerically,  or 

the  curve  lies  ivholly  idthin  the  limits  B  and  B'  along  T.  The 
form  of  the  ellipse  is  best  observed  by  the  following  mechani- 
cal construction:  Take  a  string  whose  length  is  AA'=2a,  fix 
its  extremities  at  F  and  F',  place  a  pencil  point  against  the 


80  ANALYTIC    GEOMETRY. 

string,  keeping  the  string  stretched  ;  as  the  pencil  is  moved  it 
will  trace  the  ellipse,  for  in  all  its  positions  FP-\-  PF'=  2  a. 

55.  Defs.     AA'  is  called  the  transverse  axis  of  the  ellipse, 

BB'  the  conjugate  axis,  A  and  A'  the  vertices,  FA  and  FA'  (or 

F'A  and  F'A')  the  focal  distances,  the  double  ordinate  through 

either  focus,  as  GG\  the  parameter,  and  the  distance  from  the 

/  Ff)\ 
focus  to  the  centre  divided  by  the  semi-transverse  axis  (  —  i 

the  eccentricity.  As  referred  to  an  origin  at  its  centre  and 
axes  of  reference  coincident  with  those  of  the  ellipse,  Eq.  (1), 
Art.  54,  is  called  the  central  equation  of  the  ellipse. 

56.  Common  form  of  the  central  equation.     Representing  the 

FO      c 
eccentricity  by  e,  we  have  e  — =  -,  .-.  c  =  ae,  which  substi- 

•^     -^  AO      a 

tuted  in  Eq.  (1),  Art.  54,  gives 

another  form  of  the  central  equation,  in  terms  of  the  eccen- 
tricity. Representing  the  conjugate  axis  BB'  by  2&,  26  = 
2Va^— C-,  .-.  a^—c'^=lr,  which  substituted  in  Eq.  (1),  Art. 
54,  gives  ay+6-.r=fr&-,  (2) 

the  equation  of  the  ellipse  in  terms  of  the  semi-axes,  and  called 
the  common  form  of  the  central  equation. 

Cor.  1 .    Since  e  =  -  and  a  >  c,  the  eccentricity  of  the  ellipse  is 
a 

alioays  less  than  unity. 

CoR.   2.     Since  c-=  cfi—b',    e  =  -  = ,  the  eccentricity 

.    .  ^  .,  .  a  a 

in  terms  of  the  semi-axes. 

Cor.  3.  Since  e  =  -,  c  =  oe,  the  distance  of  either  focus  from 
the  centre. 

Note.  The  Btiident  will  observe  Uiat  Uie  form  of  the  ellipse  will  vary  with  a,  b,  c, 
Bnd  e,  and  therefore  that  the  constants  in  the  equation  of  a  locus  may  serve  to  determine 
Us  form  as  well  as  its  magnitude  and  position  (Art.  16). 


COMMON   EQUATIONS    OF    CONIC    SECTIONS. 


81 


57.  Length  of  the  focal  radii. 

F  being  any  point  of  tlie  ellipse  (Fig.  45), 

FP2=  F]\P^MP^=  {ae  +  a;)-+  f  (Art.  50,  Cor.  3) 
^{ae  +  x)-+{a--x'){l-e-)  (Art.  5G,  Eq.  1) 
=  (r  -\-  2  aex  +e-a;"  =  (a  +  ex)- ; 
or  FP  =  «  +  ex. 

But        FP  +FP  =2  a,  .  • .  F'P  =  2  a  -  (a  +  ex)  =  a-  ex. 
Hence,  the  focal  radii  to  any  point  ivliose  abscissa  is  x  are  a  ±  ex. 

58.  Polar  equation  of  the  ellipse. 

Let  the  pole  be  taken  at  the  left-hand  focus,  and  the  polar 
axis  coincident  with  the  transverse  axis.  We  shall  obtain  the 
equation  directly  from  the  figure,  this  being  easier  than  to 
transform  the  central  equation.  From  the  triangle  FPF',  P 
being  any  point  of  the  curve, 

F'P'  =  FP-  +  FF'-  -2FP.FF'  cos  F'FP. 

But  FP=r,  F'FP=  6,  FF'  =  2ae,  and  F'P  =  2a-FP=^2a-r. 
Making  these  substitutions,  we  obtain 


ail-e-) 
1  —  e  cos  6 


(1) 


Discussion  of  the  Equation.     "When 

^=0°,  r  =  a{l  +e)  =  FA'; 

when  $  =  180° ,  r  =  a{l  —  e)  =  FA  ;  hence  the  focal  distances  are 
0(1  ±e). 


t~  '^'■' 


82  ANALYTIC    GEOMETRY. 

AVhen  0  =  90%  r  =  a  ( 1  -  e-)  =  a/l  -  '^\  =  a^^^^  =  -  ; 

\        a-J  a^         a 

hence  the  parameter  GG'  =  2a(l  —  e-),  or  ^^• 

a 

When  »  =  i^'i^i3  =  008^^;^=  COS"  ^ — i    r= r-i 

FB  r  ae- 

r 
.-.  r  =  a=FB.     This  is  also  evident  from  the  right-angled  tri- 
angle FOB,  in  which  FO  =  c,  OB  —  b,  and  therefore 

FB  =  V  c^TP  =  a, 

since  a-  —  c^  =  &-.  Hence  ^/ie  distance  from  either  focus  to  the 
extremity  of  the  conjugate  axis  is  equal  to  the  semi-transverse  axis. 
Therefore,  to  find  the  foci  lohen  the  axes  are  given,  with  the 
extremity  of  the  conjugate  axis  as  a  centre  and  the  semi-trans- 
verse axis  as  a  raditis  describe  an  arc;  it  ivill  cut  the  transverse 
axis  in  the  foci. 

Representing    the   parameter   GG'  =  2a  (I  —  e^)  by  2^9,  the 
polar  equation  (1)  may  be  written 


}•  = 


P 


1  —e  cos  6 


(2) 


59.  The  ratio.  The  ellipse  can  be  traced  by  a  point  so  mov- 
ing that  the  ratio  of  its  distances  from  a  fixed  x>oint  and  a  fixed 
straight  line  is  constant. 

From  the  polar  equation  r  =  :, p.,    we  have 

'■  1  —  e  cos  d 

r  =  p  -\-  er  cos  6, 
e  . 

or  FP  =  J) -\-i  FM  (Fig.  46) .     Take  FS  such  that  FS  =  -,  or 
J)  =  eF/lS,  and  draw  DD'  perpendicular  to  FS.     Then 

FP=e  (FS  +  FM)  =  eSM=  ePQ, 

FP 

PQ  being  parallel  to  MS.     But  e  is  a  constant  •  hence  — —  is  a 

constant.  ^ 

The  fixed  line  DD'  is  called  the  Directrix. 


COMMON    EQUATIONS   OF   CONIC    SECTIONS.  83 

Cou.   1 .      The  ratio  is  equal  to  the  eccentricity  and  is  always 
less  than  unity. 

IF 
CoK.  2.      Since  ^  is  a  point  of  the  curve,  - — ^  =  e, 

.  cr      -4^      a(\—e)    ,.    .    ~o\ 
.'.AS=         =-^ ^  (Art.  o8). 

e  e 

For  tlie  same  reason  - —  =  <?,  .-.  A'S  — =  -^ —       ^  •  Hence, 

A'S  e  e 

the  distances  from  the  vertices  to  the  directrix  are  —^ — — — ^• 

e 

Cor.  3.      FS  =  FA  +  AS  =  a{l  -  e)  ^  ot(l-4^  aCl-e'^)^ 

the  distance  from  the  focus  to  the  directrix. 

CoK.  4.      OS  =  0F+  FS  =  ae  +  ^''^^~  ^'^  =  -,    the  distance 
from  the  centre  to  the  directrix.  .   . 

60.    Geometrical  construction  of  the  ellipse  tvhen  the  ratio  is 
given. 

Lete  =  -,  in  which  A;<s,  be  the  given  ratio.     Take  SF=s, 
s 

draw  GG'  perpendicular  to  SF,  and  make  FG  =  FG'  =  k. 
Draw  SG  and  SG',  and  between  these  lines  produced  draw  any 
parallel  to  GG',  as  N'L'.  With  F  as  a  centre  and  3I'N'  as  a 
radius  describe  an  arc  cutting  the  parallel  in  P'  and  P" ;  these 
are  points  of  the  ellipse.  To  prove  that  P'  is  a  point  of  the 
ellipse,  draw  DD'  perpendicular  to  SF  through  S,  and  P'Q' 
])arallel  to  FS.     Then,  from  similar  triangles, 

M'M':3I'S::GF:FS; 

but  N'lF  =  FP\  M'S  =  P'Q'; 

hence  FP' :  P'Q' : :  GF:  FS, 

FP'       GF 
or  = =  e. 

P'Q'      FS 
In  the  same  way  any  number  of  points  may  be  constructed. 


84 


ANALYTIC   GEOMETIIY. 


It  is  evident  from  the  construction  that  SG  and  SG'  can  have 
but  one  point  each  in  common  with  the  curve  ;  for  this  reason 


_^^ 


Fig.  47. 


they  are  called  the  focal  tangents,  and  since  GF<FS,  their 

included  angle  is  less  than  90°.     So  long  as  e  =  -  remains  the 

s 

same,  the  distance  SF,  taken  to  repi'esent  s,  simply  determines 

the  scale  to  which  the  ellipse  is  constructed,  the  angle  between 

the  focal  tangents  remaining  the  same.     But  if  e  varies,  the 

angle  GSG'  will  vary,  and  the  ellipse  will  change  in  shape  as 

well  as  in  size. 

A  pi 

Cor.  1.     Since  -<4  is  a  point  of  the  curve  — —  =  e.     But 

^=e,.-.AF=AK. 
AS 


COMMON    EQUATIONS    OF   CONIC    SECTIONS.  85 

Similarly  A'F=  A'K'.  Hence,  to  find  the  focal  tangents  ivken 
the  axes  are  given,  first  determine  the  focus  (Art.  58)  F;  then 
make  AK=  AF  and  A'K'  =  A'F;  KK'  will  be  the  focal  tan- 
gent, and  its  intersection  with  the  axis  produced  (6')  a  point  of 
the  directrix. 

CoK.  2.     From  Geometry, 

OE  =  \{AK+  A'K')  =  \{AF+FA')  =  a. 

Hence  FB=  OE  =  a,  as  already  shown. 

Cor.  3.  Since  the  curve  is  symmetrical  with  respect  to  its 
axes,  and  OF  =  OF',  there  is  another  directrix  on  the  right  of 
the  centre  at  the  same  distance  from  it  as  DD'. 

.■  -^ 

61.  The  circle  is  a  particular  case  of  the  elhpse. 

Making  a=  h  in  the  equation  of  the  ellipse  a-y^  +  b^xP  =  a^h^, 
we  have  y-  +  ^'  =  «",  which  is  the  central  equation  of  the  circle 
whose  radius  is  a  (Eq.  (2),  Art.  47). 

Cor.  1.  When  a  =  6,  e  =  — ^ =  0;  hence  the  eccen- 
tricity of  the  circle  is  zero. 

CoR.  2.  When  a  =  b,  c-  =  cr  —  b'-  =  0  ;  hence  the  foci  of  the 
circle  are-at  the  centre. 

CoR.  3.     Since,  for  the  circle,   e  =  0,      =  the  distance  from 

e 

the  centre  to  the  directrix  =  x  ;  hence  the  directrix  of  the  circle 
is  at  infinity  and  the  focal  tangents  parallel, 

62.  Varieties  of  the  ellipse. 

Every  equation  of  the  form 

^lr+6V+F=0,  (1) 

ichich  is  not  imj^ossible,  is  the  centred  equation  of  an  ellipse,  if  A 
and  C  have  like  signs  and  neither  is  zero. 

First.  Let  i^  be  negative.  Then  ^1/H  Cx?  =  F.  If  this  is 
the  central  equation  of  an  ellipse,  it  is  reducible  to  the  form 


86  ANALYTIC    GEOMETllY. 

a-rf-\-  b-x-  —  crb'-,  iu  which  the  absolute  terra  is  the  product  of 
the  coefficients  of  the  squares  of  x  aucl  y.     Let  li  be  the  factor 

which  renders  RA  .  RC  =  RF.      Then   7?  = Introduciuo- 

An  ^ 

this  factor,  (1)  becomes  — 2/-  +  — ar=  — — ,  which  is  the  required 

form,  and  in  which,  therefore,  a=-^—^  b  =\j—  are  the  semi- 

axes.    If  -4  =  (7,  the  axes  become  equal,  and  the  ellipse  becomes 
a  circle,  which  we  have  seen  is  a  particular  case  of  an  ellipse. 

Second.  If  F=  0,  (1)  becomes  Ay--{-  Caf=:  0,  which  is  true 
only  for  .^•=0,  ?/=0,  and  the  ellipse  becomes  a  point,  which, 
as  the  limiting  case  of  a  circle,  may  also  be  considered  a  variety 
of  the  ellipse. 

Third.  If  F  is  positive,  (1)  becomes  Ay^+  Cx^  =  —  F,  which 
is  impossible,  as  the  first  number  is  the  sum  of  two  squares.    In 

this  case  2/=\| —. 5  which  is  imaginary  for  all  values  of 

ic,  as  are  also  the  semi-axes  \\ and  \ 

\    C  ^  A 

There  are  then  four  varieties  of  the  ellipse,  in  which  the  axes 

are  real  and  unequal,  real  and  equal,  zero,  or  imaginary ;  and 

we  may  say  that  every  equation  of  the  form  Ay--\-Cx"-\-F=  0, 

in  tohich  A  and  G  have  like  signs,  is  the  equation  of  an  elli2)se,  real 

or  imaginary,  according  as  F  is  negative  or  positive. 

Examples.   1.  AVhat  are  the  axes  of  the  ellipse  t)y"+  60;-=  20? 

F      10 
Multiplying  by  the  factor  It  — =  —_,  tlio  equation  becomes 

-{'- y'^  + -^  x"^  —  ^i~  ,  wliich  is  in  tlie  form  a^y^  ■\-  b^x^  =  a'^b^, 

and  we  see  by  inspection  that  the  axes  are  2  V^  and  2V^.  Otherwise, 
since  the  equation  is  the  central  equation,  the  semi-axes -are  the  intercepts, 
and  we  may  find  them  directly  by  making  .r  =  0,  .•.  y  =  6=v^,  and 
y  =  0,  .-.  x=n=  -s/J^-. 

2.    Find  the  axes,  eccentricity,  and  parameter  of  3  y-  +  'iy?=i  18, 

Avs.    o  =  3  ;  ?>  =  VC) ;  e  =  —  ;  'Ip  =  4. 


COJNOIOX    EQUATIONS    OF    CONIC    SECTIONS.  87 

3.  Find  the  axes,  focal  distances,  and  distance  of  the  direc- 
trix from  tlie  centre  of  6//-+  ^.r  =  108. 

Ans.    a=G;  b  =  SV^  ;  3V2(V2  ±  1);   gV^. 

4.  Write  the  equation  of  the  ellipse  whose  axes  are  18  and  10. 

Ans.    8\y-+26x'  =  2025. 

5.  AVrite  the  equation  of  the  ellipse  whose  eccentricity  is  | 
and  transverse  axis  10.  Ans.    9  >/--{- ox' =  125. 

G.    The  conjugate  axis  of  an  elUpse  is  4  and  its  lesser  focal 
distance  1.     Find  its  eccentricity  and  equation. 

Ans.    f  ;   257/'+  IGx-  =  100. 

7.  Construct  geometrically  the  eUipse  in  the  following  cases  : 

(a)  e=s.     Observe  the  size  is  undetermined. 

(b)  f  =  i  ;  distance  from  focus  to  directrix  =10. 

(c)  a  =  5,   h  =  4. 

8.  The  eccenti'icity  of  an  ellipse  being  — -,  what  is  the  angle 
between  the  focal  tangents  ?  ^^  Ans.    60°. 

9.  Find  the  focal  distances,  conjugate  axis,  parameter,  and 

position  of  the  directrix,  of  tlie  ellipse  r  = -• 

^  16-9cos6' 

Ans.    4,  11 ;  f  Vt  ?   2?  4^i  from  centre. 

10.  AYrite  the  polar  equation  of  the  ellipse  whose  axes  are  18 

-^^8-  Ans.    r=  ''' 


9  —  V65  cos^ 
THE    HYPERBOLA. 

63.  Defs.  The  path  of  a  poi)d  so  moving  that  the  difference 
of  its  distances  from  tico  fixed  points  is  constant  is  called  an 
hyperbola.  The  two  fixed  points  are  called  the  foci,  the  point 
midway  between  them  the  centre,  and  the  lines  joining  any 
point  of  the  hyperbola  with  the  foci  the  focal  radii. 

64.  Central  equation  of  the  hyperbola. 

Let  F,  F',  be  the  foci,  0  the  centre  and  origin,  the  axis  of  X 


88  ANALYTIC    GEOMETKY. 

being   coincident   with  FF\   P  any   point  of   the   hyperbola, 
FF'  =  2  c,  and  2  a  the  constant  difference.  Then  FP-F'P=  2  a. 


But  FP  =  Vi^J/-  +  MP'  =  V  (x-  +  c)  -  +  f, 


F'P  =  ■VF'3P+  MP-  =  V  {X  -cy+  y\ 


Hence      V(a;  4-c)^+r— V(.i- —  c)-+ ?/- =  2a. 

Transposing  the  second  term  to  tlie  second  member,  and 
squarmg,     ^^^  ^  ^_^,,^  ^^^,  ^  ^  ^^,^  ^ ^^  ^______,^  (.^_e)2+  ^.^ 

or  ex  —  a-  —  a  V  (if  —  c)  ^+  y^- 

Squaring  again, 

ay  4-  (a2  -  C-)  .r-'  =  (r{a'  -c').  '  ( 1 ) 

Discussion  of  the  equation.  Since  only  the  squares  of  the 
variables  enter  the  equation,  the  hyperbola  is  symmetrical  with 
respect  to  both  axes.  Making  y  =  0,  the  X-intercepts  are  ±  o. 
Take  OA=OA'=a,  then  yL4'=2cf,  the  constant  diflerence. 
and  as  the  difference  between  two  sides  of  a  triangle  is  less  than 
the  third  side,  PF  -  PF' =  2a<  FF' =  2r,  or  A  and  A'  are 
betzceen  the  foci.  Making  x  =  0,  the  F-intercepts  are  ±Va'— o-, 
and  are  imaginary,  as  a  <  c  ;  hence  the  curve  does  not  cross  the 
axis  of  Y.     Solving  the  equation  for  »/, 

1      

wliich,  since  cC-—(?  is  negative,  is  imaginary  for  all  values  of 
a:  <  a  numerically,  and  the  curve  lies  icholly  tdthoiit  the  limits  A 
and  A'  along  X,  extending  to  ±  cc.     Solving  for  x, 


±a^ii — r_ 

\       a^-  c- 

which  is  real  for  all  values  of  y  since  a-—  c-  is  negative,  or  the 
curve  has  no  limits  in  the  direction  of  Y.  The  form  of  the  hyper- 
bola is  best  observed  by  the  following  mechanical  construction  : 
take  a  ruler  of  any  lengtli  Fl.  (ixinu-  one  extremity  at  F,  and  a 
string  F'PI,  shorter  than  the  ruler  by  .Ll'=  2a,  one  of  whose 


COMMON    EQUATIONS    OF    CONIC    SECTIONS. 


89 


ends  is  attached  to  the  farther  extremity  /  of  the  ruler,  the 
otlicr  at  the  focus  F'.  Press  the  string  against  tlic  ruler  l)^'  a 
pencil,  as  at  P,  keeping  the  string  stretched,  the  ruler  turning 


Fig.  48. 

about  its  fixed  end  F.  As  the  pencil  moves  it  will  trace  the 
hyperbola,  for  in  all  its  positions  FPI  =  F' PI -\- 2  a  ;  or,  sub- 
tracting PI  from  each  number, 

FP  =  F'P  +  2  a,    .  • .  FP  -  F'P  =  2  a. 

65.    Defs.     AA'  is  called  the  transverse  axis  of  the  hyper- 

bola,  A  and  A'  the  vertices,  FA  and  Fxi'  (or  FA'  and  F'A)  the 

focal  distances,  the  double  ordinate  through  either  focus,   as 

(?(?',  the  parameter,  and  the  distance  from  the   focus  to  the 

/F0\ 
centre  divided  by  the  semi-transverse  axis  f |,  the  eccentric- 

•^  \AOj 

ity.  Equation  (1),  Art.  64,  is  called  the  central  equation  of 
the  hyperbola. 


66.  Common  form  of  the  central  equation.  The  hyperbola 
does  not  cross  the  axis  of  Y,  and  does  not  therefore  determine 
by  its  intercepts  a  conjugate  axis,  as  in  the  case  of  the  ellipse. 
Its  equation,  however,  will  assume  a  form  similar  to  that  of  the 

ellipse   if  we  represent  the  viirnerical  value  of  V«"  —  c'  by  6, 
laying  off  OB  =  OB'  =  b  (Fig.  4.S)  for  a  conjugate  axis.    We  thus 


90  ANALYTIC    GEOIVIETRY. 

have  «-—  c-  =  —  b'-,  minus  because  a<:,c,  aud  e  =  -,    .-.  c  =  ae, 
e  being  the  eccentricity. 

Substituting  c  =  ae  in  Eq.  (1),  Art.  64,  we  have 

or,  since  o  >  a,  and  therefore  e  =  -  >  1, 

a 

the  central  equation  in  terms  of  the  eccentricit\'.     Substituting 
a^—c-  —  —  b-,  in  the  same  equation,  we  obtain 

cr  y-  —h-!xr  =  —  tr  6",  (  2  ) 

the  common  form  of  the  central  equation  of  the  liyperbola. 

Note.  The  student  will  observe  that  Equations  (1)  and  (2)  diflter  from  the  corre- 
sponding equations  of  the  ellipse  (Art.  56)  only  in  the  value  of  e  and  the  sign  ofb'-;  also, 
that  while  .r=0,  in  Eq.  (2),  gives  2/=  ^  V—  li'-'  ^f  imaginary  quantity  (as  it  should 
be,  since  the  curve  does  not  cross  Y),  its  numerical  value  is  the  semi-conjugate  axis,  as 
in  the  case  of  the  ellipse. 

Cou.  1.  Since  ^=    ,  aud  e>o,  tlie  eccentricity  of  the  luiper- 
a 

bola  is  cdicays  greater  than  unity. 

Cor.  2.  Since  a^—  c-  =  — ir,  e  =  ~  = — — ,  the  eccentricity 

in  terms  of  the  semi-axes. 


Cor.  3.    Since  e  =  -,    c  =  ae=  ■\/a:-'+b-  =  AB  (Fig.  48), 


or 


a 

the  distance  from  the  focus  to  the  centre  is  the  distance  from  either 
vertex  to  the  extremity  of  the  conjugate  axis.  Hence,  to  find  the 
foci  ichen  the  axes  are  given,  v:ith  0  as  a  centre  and  AB  as  a 
radius,  describe  an  arc;  it  iviJl  cut  the  transverse  axis  in  the  foci. 

67.    Length  of  the  focal  radii.       P  l)eing  any  point  of  the 
hyperbola  (Fig.  48), 

FP''  =  FM-  +  MP-  =  {ae  +  x)-+y-  (Art.  G6,  Cor.  3) 
=  (ae  +  a;)2+(ar^-fr)(e'-  0  (Art.  GG,  Eq.  (1)) 
=e'^x'^+2aex  +  a^  =  {ex  +  af  ; 
or       FP  =  ex  -\-  a . 


COMMON    EQUATIONS    OF    CONIC    SECTIONS. 


91 


But  F'P  =  Fr-'2 a  =  ex-\-a—2a  =  ex  -  <(. 

Hence  the  focal  radii  to  any  point  ivhose  abscissa  is  x  are  ex  ±  a. 

68.    Polar  equation  of  the  hyperbola. 

Let  the  pole  be  taken  at  the  left-hand  focus,  and  the  polar 
axis  coincident  with  the  transverse  axis.  From  tlie  triangle 
FPF',  P  being  an}-  point  of  the  curve, 

F'P^  =FP^+  FF''—2FP .  FF' cos F'FP. 

But  FP=r,  F'FP  =  9,  FF'=^2ae,  and  F'P=FP-2a  =  r-2a. 
Making  these  substitutions,  we  obtain 

a(e'—l) 


r  = 


e  cos  ^  —  1 


(1) 


Discussion  of  the  equation'.  When  ^=0°,  r=a{e-{-l)=FA'; 
when  6  =  180°,  r  =  —a{e  —  l)  =  FA  ;  or  the  focal  distances 
are  a  {e  ±  1),  numerically. 


1    = 


b' 


a 


;  or    the      u 


Wlien  ^  =  90°,  r  =  -  a  (e--\)  =  -  a 

parameter,  GG',  is  2a  (e'  —  l),  or  ^^-,  numerically. 

As  6  increases    from   0°,    cos (9  diminishes    and  r  increases, 
tracing  the  branch  ^1'  P,  r  becoming  infinity  when  e  cos  ^  =  1 , 


Fig,  49. 


92  ANALYTIC   GEOMETRY. 

or  ^  =  cos"^--     When  ecos^^l,  r  is  negative,  and  the  branch 

e 

G'A  is  traced,  in  the  direction  G'A,  r  being  FA  when  ^  =  180°. 
"When  0  passes  180°,  cos^  is  negative  and  /•  remains  negative, 
tracing   the   branch   AG,    and    becomes    infinity    again    when 

ecos^  =  l,    or   ^=cos^^-    in   the  fourth   angle;    after  which, 

e 

ecosO  is  greater  than  unity,  /•  is  positive  and  ti'aces  the  branch 
LA'. 

Representing   the    parameter   GG'  —  2a{e-  —  l)    by    2p-,  the 
polar  equation  (1)  may  be  written 


P 


e  cos  0  —  1 


(2) 


69.  The  ratio.  The  hyperbola  can  be  traced  by  a  xioint  so 
moving  that  the  ratio  of  its  distances  from  a  fixed  point  and  a 
fixed  straight  line  is  constant. 

From  tlie  polar  equation  of  the  hyperbola,  r= -^ ,  we 

ecos^  — 1 

/       have  r  =  ercose-p,  or  (Fig.  49)  FP=eF3I-p.     Take  FS 

P 
such  that  FS  =    ,  or  p  —  eFS,  and  draw  DD'  perpendicular  to 

FS.     Then  FP=e  {FM-  FS)  =  e  SM=  e  PQ,  PQ  being  par- 

FP  . 
allel  to  MS.     But  e  is  a  constant ;  hence  — -  is  a  constant. 

Pil 

The  fixed  line  DD'  is  called  the  directrix. 

Cou.  1.    The  ratio  is  equal  to  the  eccentricity,  and  is  always 
greater  than  unity. 

Cor.  2.  Since  A  is  a  point  of  the  curve, 

^^=e,  .•.^5  =  -i^="^'^-^^  (Art.  68V 
AS  e  e 

For  the  same  reason  ^^e,  .-.  A'S  =  —  =  'ii^-tD.    Hence 

A' S  e  e 

the  distances  f roni  the  vertices  to  the  directrix  are  —^ — ~ — ^« 


COMMON   EQUATIONS    OF   CONIC    SECTIONS.  93 

Cor.  3.    FS  =  FA  +AS  =a  (e  - 1)  +  ^  ^^  ~  ^ ^  =  ^  C^'-^)  ^ 

e  e 

the  distance  from  the  focus  to  the  directrix. 

Cor.  4.    OS  =  0F- FS  =  ae -"'^^'~^^  =-=   the  distance 

e  (' 

from  the  centre  to  the  directrix.  /  ,-) y  >  ^ 

70.    Geometrical  construction  of  the  hyperbola  when  the  ratio 
is  given. 

Let  e  =  -,  in  which  A;  >  s,  be  the  given  ratio.     Take  FS  =  s, 
s 

draw  GG'  perpendicular  to  FS,  and  make  FG  =  FG'  =  k. 
Draw  SG  and  SG',  and  between  these  lines  produced  draw  an}- 
parallel  to  GG',  as  N'L'.  With  i^  as  a  centre  and  M'N'  as  a 
radius,  describe  an  arc,  cutting  the  parallel  in  P'  and  P" ; 
these  are  points  of  the  hyperbola.  To  prove  that  P  is  a  point 
of  the  hyperbola,  through  S  draw  DD'  perpendicular  to  SF, 
and  P'Q'  perpendicular  to  DD' .     Then,  from  similar  triangles, 

N'M':M'S:'.GF'.FS\ 


but 

N'M'  =  FP',    M'S  =  P'Q' ; 

hence 

FP'-.P'Q'::  GF:FS, 

or 

FP'      GF 
P'Q'      FS 

Since  iViJij  >  MiS  b}'  construction,  the  arc  described  with 
FPi=M\Ni,  as  a  radius  will  determine  Pj,  P^,  on  the  right  of 
DD',  which  may  be  proved  to  be  points  of  the  hyperbola  as 
above  ;  and  in  the  same  manner  any  number  of  points  may  be 
constructed. 

It  is  evident  from  the  construction  that  SG  and  SG'  can  have 
but  one  point  each  in  common  with  the  curve  ;  for  this  reason 
they  are  called  the  focal  tangents.  Since  GF>FS,  their  in- 
cluded angle,  is  greater  than  90°,  the  distance  FS,  taken  to 
represent  ^.  simply  determines  the  scale  of  the  construction ; 
but  if  e  varies,  the  angle  G'SG  will  vary,  and  the  hyperbola  will 
differ  in  shape  as  well  as  size. 


94 


ANALYTIC    GEOMETRY. 


Cor.   1.    Since   ^1   is   a   point  of  the   curve,   — —  =  e.    But 

^—=e,..AF=AK.      Similarly,    A'K'=A'F.      Hence,    to 
AS 

find  the  focal  tangents  ivhen  the  axes  are  given,  first  determine 


Fig.  50. 


the  focus  (Art.  66,  Cor.  3),  thou  make  AK=AF  and 
A'K'=  A'F.  KK'  will  be  the  focal  tangent,  and  its  intersec- 
tion with  the  axis,  S^  a  point  of  the  directrix. 


COMMON   EQUATIONS    OF   OONIC    SECTIONS.  95 

Cor.  2.  Since  the  curve  is  symmetrical  with  respect  to  its 
axes,  aud  CF'  =  CF,  there  is  another  directrix  ou  the  right  of 
the  centre  at  the  same  distance  from  it  as  DD'. 

71.  The  equilateral,  and  the  conjugate  hyperbola. 

When  the  axes  of  an  hyperbola  are  equal,  it  is  said  to  he  equi- 
lateral. ]M:iking  a  =  h  m  the  common  form  of  tlie  central  equa- 
tion a-y-  —  b"x-  =  —  a~b~,  we  have 

y--x-  =  -a\  (1) 

for  the  equation  of  the  equilateral  hyperbola. 

Tivo  hyperbolas  are  said  to  be  conjugate  to  each  other  when  the 
transverse  and  conjugate  axes  of  the  one  are  the  conjugate 
and  transverse  axes  of  the  other.  If,  in  deducing  the  equation 
of  the  h3perbola,  the  transverse  axis  had  been  assumed  coinci- 
dent with  y,  tlie  equation  of  the  hyperbola  in  this  position 
would  have  been  a-x^  —  b^y^  =  —  a-b'',  as  this  supposition  simply 
amounts  to  interchanging  x  and  y.  Interchanging  now  a  and 
?>,  this  becomes  ^^,^,  _  ^,^  ^  ^^,^, .  ^g) 

or,  the  central  equations  of  conjugate  hyperbolas  differ  only  in  the 
sign  of  the  absolute  term. 

Conjugate  hyperbolas  are  distinguished  as  the  X-  and  the  T- 
hyperbola,  each  taking  its  name  from  the  coordinate  axis  on 
which  its  transverse  axis  lies,  and  the  equation  of  either  may 
be  derived  from  that  of  the  other  by  changing  the  signs  ofa^ 
and  W. 

Cor.  1 .  The  eccentricity  of  the  ^-hyperbola  is  — — 

Cor.  2.  Since  the  distance  of  the  foci  of  an  hyperbola  from 
the  centre  is  the  distance  between  the  extremities  of  the  axes 
(Art.  66,  Cor.  3),  the  foxir  foci  of  a  pair  of  conjugate  hyperbolas 
are  equidistarit  from  the  ceyitre. 

72.  Varieties  of  the  hyperbola.     Every  equation  of  the  form 

J/+ar  +  F=0  (1) 


96  ANALYTIC    GEOMETRY. 

is  the  central  equation  of  an  hyperbola,  if  A  and  C  have  unlike 
signs  and  neither  is  zero. 

First.    Let  F  be    positive.      Then   Ai/  —  Cxr  =  —F,    which 
can  be  reduced  to  the  form  a-y-  —  Irx-  =  —  d-b'-,  as  in  the  case 

F 

of  the  ellipse   (Art.  62),  by  introducing  the  factor  B  =  -—  ; 


p  rp  pi2  IP  I  Jp 

whence  ~y- x^  = -,  in  which  a  =  ^  — -,  and  b  =\l — —- 

C         A  AC  \  C  y   A 

numerically.  If  A  —  C,  the  axes  are  equal  and  the  hyperbola 
is  equilateral. 

Second.  If  i<'is  negative,  (1)  becomes  Ay^  —  Cx^'  =  F,  which 
is  the  conjugate  hyperbola,  since  it  differs  from  Ay-  —  Cxr  =  —  F 
only  in  the  sign  of  the  absolute  term  (Art  71). 

fn 
Third.    If  F  =  {),  (1)  becomes  At/  —  Cxr  =  0,  or  y=±  \-7X, 

which  is  the  equation  of  two  straight  lines  through  the  origin 
making  supplementary  angles  with  X.  In  this  case  the  axes 
are  zero. 

There  are  then  four  varieties  of  the  hyperbola,  in  which  the 
axes  are  unequal,  equal,  interchanged,  and  zero;  corresponding 
to  the  X-,  equilateral,  Y-hyperbola,  and  a  pair  of  intersecting 
straight  lines  through  the  origin. 

Examples.     1.  What  are  the  axes  of  the  hyperbola 

9y--4.x-  =  -Ui? 

Multiplying  by  the  factor  R  =  -—  =  4,  the  equation  becomes 

AC 

36  3/2  -  16  x^  =  —  576,  which  is  of  the  form  a^  y^  —  b-  x-  =  —  a-  b- ; 

tlie  axes  are  therefore  12  and  8.  Or,  directly,  making  y  =  0  and  x  =  0  in 
succession,  wo  have  numerically  .r  =  a  =  6,  ?/  =  6  =  4,  .-.  2  a  =  12,  2  6  =r  8. 

2.  Find  the  axes,  eccentricity,  and  parameter  of 

3?/2_2.r2  =  -18. 

Ans.  6,  2V6;  iVT5;  4. 


COMMON    EQUATIONS    OF   CONIC    SECTIONS.  97 

3.  Find  the  axes,  focal  distances,  and  position  of  the  direc- 
trix of  ?/-a-2  = —81.  y 

Ans.  a  =  b=0  ;  0( V2  ±  1)  ;  ~r:  from  centre. 

V2 

4.  Write  the  equation  of  the  hyperbola  whose  axes  are  18  and 
10.  Ans.  81  ?/2- 25  a- =  -2025. 

5.  AVrite  the  equation  of  the  hyperbola  whose  eccentricity  is  |- 
and  transverse  axis  10.  Ans.   D  y-  —  1  xr  =  —  175. 

G.  The  conjugate    axis  of    an  hyperbola  is  4  and  its  lesser 
focal  distance  1.     Find  its  eccentricity  and  write  its  equation. 

Ans.  |;  9?/--16.'>r  =  -36. 

7.  Construct  the  following  hyperbolas  : 

(a)   e  =  f .  Observe  the  size  is  undetermined. 
(^b)  e  =  f  ;  distance  from  focus  to  directrix  =  8. 
(o)   a  =8,  b  =  6. 

8.  Construct  a  pair  of  conjugate  hyperbolas  whose  axes  are 
12  and  8. 

9.  Write  the  equations  of  the  hyperbolas  conjugate  to  those  of 
Exs.  2  and  3,  and  determine  their  eccentricities  and  directrices. 


Ans.  < 


3  r-  2  X-  =  18  ;    ^;^- ;    2  J-  from  centr 

9 
i/  —  af=81  ;   V2  ;  —7=  from  centre. 

V2 


e. 


10.  The  eccentricity  of  an  hyperbola  being  V3,  what  is  the 
angle  between  the  focal  tangents  ?  Ans.   120°. 

11.  Find  the  focal  distances,  conjugate  axis,  parameter  and 

6 


directrix  of  the  hyperbola  r  = 


Vl5  cos^-3 


Ans.   —^ ;  2  Ve  ;  4  ;  3^  -  from  centre. 

Vl5q:3  ^5 

12.  Write  the  polar  equation  of  the  hyperbola  whose  axes  are 

8  and  6.  ^9 

Ans.  r  = -. 

5  cos  0—4: 


98 


ANALYTIC    GEOMETRY. 


THE    PARABOLA. 

73.  Defs.  The  path  of  a  point  so  moving  that  its  distance 
from  a  fixed  p)oiHt  is  always  equal  to  its  distance  from  a  fixed 
straight  line  is  called  a  parabola.  The  fixed  poiut  is  the  focus, 
the  fixed  straight  Hue  the  directrix,  and  the  line  joining  any 
point  of  the  parabola  with  the  focus,  the  focal  radius. 

74.  Equation  of  the  parabola. 

Let  F  be  the  focus,  DD'  the  directrix.  Draw  iSi^  perpendicu- 
lar to  DD'  and  let  /SF  =  2i-  By  definitiou,  the  middle  poiut  0 
of  SF  is  a  point  of  the  curve.  Let  0  be  the  origin  and 
the  axis  of  X  coincident  with  OF.  Then,  P  being  any  point 
of  the  curve,  and  PQ  perpendicular  to  DD',  PF=  PQ,  or 

P 


<-'^' 


Squaring  and  reducing, 


+  r  =  x-  + 

y-  =  2X)X. 


(1) 


Discussion   of   the    equation.       Solving   for   ?/,    we    have 


y  =  ±  V2pa;,  or  y  has  two  numerically   equal  and  increasing 
values  for  positive   increasing   values   of  x,  but  is  imaginary 

when  X  is  negative ;  hence  the 
curve  lies  wholly  to  the  right  of  F, 
extends  to  infinity  in  the  first  and 
fourth  angles,  and  is  symmetrical 
with  respect  to  X.  The  form  of 
the  parabola  may  be  observed  from 
the  following  mechanical  construc- 
tion :  take  a  ruler  of  any  length 
QI,  and  a  string,  FPI  equal  in 
length  to  the  rnler.  Fix  one  end 
of  the  string  at  the  focus,  the  other 
at  the  extremity  /  of  the  ruler, 
and,  keeping  the  string  pressed  against  the  rnler  at  P  by  a 
pencil,  slide  the  ruler  along  the  directrix  parallel  to  SF;  the 
pencil  will  trace  the  curve,  for  in  all  its  positions  PQ  =  PF. 


Fig.  51. 


COISOrON    EQUATIONS    OF    CONIC    SECTIONS.  99 

OX  is  called  the  axis  of  tlie  parabola,  0  the  vertex,  and  the 
double  ordinate  GG'  through  the  focus  the  parameter. 

CoK.    1.    Substituting   .r=  OF  =  ^    in    Eq.     (1),    we    have 

y=^FG  =p,  or  the  jyarameter  GG'  =  2p  =  the  coefficient  of  x  in 
the  equation  of  the  curve.  Also  OS  =  0F= -}j2^  =  \GG' ;  or 
SF=FG=FG'=2y. 

Cor.  2.  FP=  QP=  SO  +  0M=  ^p  +  x;  or  the  length  of 
the  focal  radius  to  any  point  where  abscissa  is  x  is  x  +  ^p. 

75.   Polar  equation  of  the  parabola. 

Let  the  pole  be  taken  at  the  focus,  and  the  polar  axis  coinci- 
dent with  the  axis  of  the  parabola.  The  formuke  for  transfor- 
mation from  rectangular  axes  at  0  to  the  polar  system,  are 

p 

X  =  .!•„  +  r  cos  6  —  -r  +  r  cos  6., 

y  =  ?/,,  -f-  r  sin  6=  r  sin 6. 

Substituting  these  values  in  the  equation  y^  =  2px,  we  have, 

r^sm-6=2p(^^-{-rcos6\ 

or  ?*^(1  —  cos-  6)  =  })-  +  2pr  cos  0. 

Transposing, 

r-  =  ?•-  cos'^  -j-  22??'  cos^  -f  p^  _  ^j.  cos6  +py. 

Extracting  the  root  of  each  member, 

?•  =  - 2'    ovr  =  —^—^'  (1) 

1  —  cos  6  vers  6 

Let  the  student  discuss  the  equation. 
Observe  that  the  equation 

1  — ecos0 

is  the  general  polar  equation  of  the  ellipse,  circle,  hyperbola, 
and  parabola,  when  the  pole  is  at  the  focus ;  taking  the  forms 


100 


ANALYTIC    GEOMETRY. 


r  = 


P 


1  —  e  cos  6 


for  the  ellipse  (Art.  58),  that  is,  when  e<l  ;  r=R 

p 

for   the    circle  (Art.  52),   when   e  =  0  ;  r  = -^ for  the 

^  '  e  cos  0  —  1 

n 

for  the 


hyperbola   (Art.  68),    when   e>l;    and    r  =  - ^ 

parabola,  when  e  =  1 . 


Q 


76.  Geometrical  construction  of  the  parabola,  the  focus  and 
directrix,  or  the  i^arameter,  being  given. 

Lay  off  SF=2^  =  J  the  parameter,  or  the  given  distance 
between  the  focus  and  the  directrix.  Draw  GG'  perpendicular 
to  SF,  and  make  FG  =  FG'  =  SF.  Draw  SG  and  SG\  and 
any  chord  N'L'  perpendicular   to  SF.     With  i^  as  a  centre 

and  J/'A^'  as  a  radius  describe  an 
arc  cutting  the  chord  in  P'  and  P". 
These  are  points  of  the  parabola. 
To  prove  that  P'  is  a  point  of  the 
parabola,  join  P  with  P,  and  draw 
P'Q'  parallel  and  DD'  perpendicular 
to  SF.     Then 

N'M'  ^GF^  P'F 
31' S      FS      P'Q'' 

since  the  triangles  GFS  and  N'3f'S 
are  similar,  and 

P'F=  N'M',  P'Q'  =  M'S, 
In  the  same  way  any  number  of  points  may 


D 

A 

t  / 

/. 

w 

p' 

G 

/ 

\ 

F 

w 

\ 

G 

X 

X 

\^\ 

p" 

D' 

\ 

^^~-~ 

r 

\ 

Fig.  52. 


by  construction 
be  found. 

As  in  the  case  of  the  ellipse  and  the  hyperbola.  SN'  and  SL' 
have  evidently  but  one  point  each  in  common  with  the  curve, 
and  are  called  the  focal  tangents;  and  as  SF=  FG  =FG',  the 
focal  tangents  of  the  para])ola  make  an  angle  of  90°  with  each 

other.     —  =  -^-^  is  called  the  ratio,  and,  evidently,  the  ratio 
FS      P'Q' 

of  all  parabolas  is  unity. 


COMMON   EQUATIONS    OF    CONIC    SECTIONS.  101 

The  distance  SF,  taken  to  represent  p,  determines  the  scale 
to  which  the  parabohi  is  constructed.  Had  a  distance  twice 
that  of  tlie  figure  been  taken,  the  construction  of  the  same  parab- 
ola to  the  new  scale  would  have  been  equivalent  to  the  con- 
struction of  a  parabola  whose  parameter  was  2  (2p)  to  the 
original  scale.     Hence,  para^o^o.s,  like  circles,  differ  only  in  size. 

Examples.  1.  Construct  the  parabola  whose  parameter  is 
10,  and  write  its  equation. 

2.  Construct  the  parabola  the  distance  of  whose  vertex  from 
its  focus  is  2. 

3.  Write  the  polar  equations  of  the  parabolas  of  Exs.  1  and  2. 

4 

4.  The  polar  equation  of  a  i^arabola  is  r  = Write 

.^         ,       \  \.  ^  1-cos^ 

its  rectangular  equation. 

Ans.  y-  =  8x. 


/  0/ 


102 


ANALYTFC    GEOMETRY. 


SECTION    VIII.  —  GENERAL    EQUATIONS    OF   THE 
CONIC    SECTIONS. 


77.  Defs.  A  conic  is  the  locus  of  a  jwint  so  moving  that  the 
ratio  of  its  distances  from  a  fixed  point  and  a  fixed  straight  line 
is  constant.  This  constant  is  called  the  ratio,  the  fixed  point 
the  focus,  the  fixed  line  the  directrix,  and  the  [)eipendicular  to 
the  directrix  through  the  focus  the  axis  of  the  conic. 

78.  General  equation  of  the  conies. 

Let  P  be  any  point  of  the  conic,  (»i,  n)  the  focus  jP,  DD'  the 
directrix,  its  equation  being  x'lcosa  +  ?/i  sin  a  — p  =  0,  the  sub- 
scripts being  used  to  distinguish  the  coordinates  of  the  directrix 

from  those  of  the  conic.  Then  FS, 
perpendicular  to  Z>Z>',  is  the  axis. 
Join  F  with  P,  draw  PQ  perpendicu- 
lar to  DI)\  and  let  e  =  the  constant 

PF 
ratio.    Then =  e,  or 

PQ 


PF-  =  e-Pq\ 

But    PF=  VJy  -  ny  +  {x  -  m.y 
(Art.  7)  ;  and 
PQ  =  X  cos  a  +  2/  sin  a  —^)  (Art.  38)  ; 
hence       (y  —  ny +  {x  —  my  =  e-{xcosa  +  ysma—2)y         (1) 

is  the  required  equation,  in  which  e  determines  the  species,  and 
m,  ?i,  a,  and  p,  the  position  of  the  conic. 

Examples.     1.    Write  the  equation  of  an  ellipse  whose  centre 


is  (1,  2),  tran«;verse  axis  is  6,  eccentricity 
axis  parallel  to  X. 


V5 


and  transverse 


GENERAL    EQUATIONS    OF    CONIC    SECTIONS.  108 

o  —  180°,  e.  —  — -  ;   .-.  cos  a  —  —  1,  sin  a  -=  0,  p  =    "  _  —  1,    m  =  \—  Vo,  n  —  2. 
Substituting  in  Eq.  (1), 

or  i*  .'/■-  +  i  ■'■-  —  :5< ) .'/  —  8  .«•  +  4  =  0. 

2.  Write  the  equation  of  a,  parabola  whose  axis  is  parallel  to 
X,  vertex  is  at  (  —  3,  —2),  aud  parameter  is  9. 

Ans.  /  + 4?/ -9a; -23-0. 

3.  Write  the  equation  of  an  elli[)se  whose  eccentricity  is — -•> 

V3 
centre  is  (1,1),  transverse  axis  2V3,  the  latter  being  inclined 

at  an  angle  135°  with  X. 

m  =  1 -,   w  =  1  -] -,  j)  =  3,  e  =  — -,  a  =  135°, 

V2  V2  V3 

and  the  equation  is 

5  f  +  2 .17/  +  5  :r2  -  12  //  -  12  ,f  =  0. 

4.  Write  the  equation  of  a  circle  whose  radius  is  o,  the  axes 
being  tangent  to  the  circle. 

m  =  71  =  5;     If-  +  X-  -  10  //  -  10  .r  +25  =  0. 

5.  The  centre  of  an  ellipse  is  (  —  |,  4),  its  eccentricity  f,  and 
its  transverse  axis  =  J^^,  and  is  parallel  to  X ;  write  its  equation. 

Ans.   9  ?/2  +  5  a;-  -  72  ?/  -f  1 2  a;  +  144  =  0. 

79.  Every  complete  equation  of  the  second  degree  between  x 
and  y,  and  all  its  forms,  is  the  equation  of  a  conic;  and,  con- 
versel}',  the  equation  of  every  conic  is  some  form  of  the  equation 
of  the  second  degree. 

Expanding  the  general  equation  of  the  conies,  Art.  78,  we 

have 

(1  —  e-sin-a)?/^  —  2e-sina  cosa  xy  +  (1  —  ^-cos^a)  a?  ] 

-f  (2e-psina  — 2?;)?/+(2e^pcosa— 2m).T+7jr+n^— e-jr=0.  J 
The  complete  equation  of  the  second  degree  between  a?  and  y, 
Af'  +  B.ry  +  av-  +  Dy  +  Ex-\-F=0,  (2) 


104 


ANALYTIC    GEOMETRY. 


is  of  the  same  form,  but  the  coefficieats  of  corresponding  terms 
are  not  necessarily  the  same,  since  any  equation  may  be  multi- 
plied or  divided  by  any  factor  without  affecting  the  qualit}^ 
Making  these  coefficients,  therefore,  equal,  by  dividing  each 
equation  by  its  absolute  term,  and  designating  the  resulting 
coefficients  of   (2)   by  ^1',  B\  etc..  we  have 


e-  sm-a 


rfi-  +  n^  —  e^p^ 
—  2e-  sin  a  cos  a 


9      O 


=  A\ 


=  B\ 


C. 


=  D\ 


(3) 


m-+  ir 

1  O  9 

1—  e-  cos- a 

vr-\-  ir—  e-jr 

'le-p  sin  a  —  2  » 

9,9  9        9 

m--\-  n-  —  e-p' 
2  e-p  cos  a  —  2  m  _  , ,, 

9,9  9        9 

»r+  n- —  e'p- 

From  these  five  equations  the  values  of  the  five  constants 
A',  B',  C,  etc.,  may  always  be  determined  when  a,  m,  n,  p, 
and  e  are  given  ;  and  as  the  latter  are  arbitrary,  such  values 
may  be  assigned  to  them,  that  is,  the  locus  ma}'  be  assumed  of 
such  species  and  in  such  position,  as  to  give  A\  B\  C",  etc., 
any  and  every  possible  set  of  values.  Conversely,  the  values 
of  a,  ?)i,  n,  p,  and  e,  can  always  be  found  from  the  above  equa- 
tions when  those  of  A\  B\  C,  etc.,  are  given  ;  that  is,  a  conic 
of  some  species  and  position  corresponds  to  any  and  every  set 
of  values  which  may  be  assigned  to  A',  B',  C",  etc.  Hence, 
every  equation  of  a  conic  is  some  one  of  the  forms  assumed  by 
the  general  equation  of  the  second  degree^  and  every  form  of  sxich 
equation  is  the  equation  of  some  conic. 


The  axes  were  assumed  rectangular, 
have  been  (Art.  7) 


IlatI  tlicy  been  obliqnc,  the  distance  FP  would 


y/[y-ny+(,x  —  my  +  2{y-n)(x-m)  cos  /S, 

and  the  distance  PQ  would  have  been  (Art.  38) 

X  COB  a  +  y  co»p'  -  p, 

in  which  p  if  the  given  inclination  of  the  axes,  .Tnd  |3'  the  angle  made  by  PQ  with  Y. 
The  equation  P F-  =  t'-PQ-  Wduld,  therefore,  have  involveil  the  Kainc  arl)itrary  constants, 


GENEEAL    EQUATIONS    OF    CONIC    SECTIONS.  105 

and  no  others.  Passing  now  to  rectangular  axes,  since  this  trausforraation  involves  no 
new  arbitrary  constants,  and  cannot  affect  the  degree  of  the  equation,  therefore  the  above 
reasoning  is  entirely  general. 

80.    To  determine  the  species  of  a  conic  from  its  equation. 
Forming  5'-— 4J.'C"  from  Eq.  (3),  Art.  79,  we  have 

_  4 e^  sin^g  cos^g  —  4(1  —  e-  slir'a)  {I  —  e-  cos^g) 
m--\-n-—  e-iry 

_  4e^  sin^g  cos-g  —  4+46^  cos^a  +  4e^  siii'a  —  4e^  siu^g  cos^g 

nr-\-  n-—  e-p-y 
^        4(e--l) 

(m^  +  n-—  e-p'^'Y 

Now  the  locus  will  be  an  ellipse,  a  parabola,  or  an  hyperbola, 
according  as  e  is  less  than,  equal  to,  or  greater  than,  unity. 
But,  since  the  denominator  of  the  above  fraction  is  a  square, 
and  the  sign  of  the  fraction  is  thus  that  of  its  numerator,  when 
e<  1  the  first  member  is  negative,  when  e  =  1  it  is  zero,  and 
when  e  >  1  it  is  positive.  Hence  the  conic  will  be  an  ellipse, 
parabola,  or  hyperbola,  according  as  B'~  —  4^'C"  is  negative, 
zero,  or  positive. 

To  apply  this  test  it  is  unnecessary  to  reduce  the  given  equa- 
tion to  the  form'lfl^' 

A'y-+B'xy+C'x'^D'ii  +E'x  +  1  =  0; 

for  if  B''—4:A'C'  be  negative,  zero,  or  positive,  then  will 

{KB')-~  4  {KA')  {KG')  =  K-{B"~  4  A'C)    • 

also  be  negative,  zero,  or  positive.  Hence,  iv7iatever'  the  co- 
efficients, Ay-  +  Bxy  +  Cxr  -\-  Dy  +  Ex  +  F  =  0  is  the  equation 
of  an  ellipse,  parabola,  or  hyperbola,  according  as  B-—AAC  is 
negative,  zero,  or  positive. 

Examples.     Determine  the  species  of  the  following  conies : 

(1)  y-—  bxy  +  6ic^  —  14ic  +  by-\-A  =  0,  an  hyperbola; 

(2)  ?/--  8 xy  +  2b a?  +  6y  -  2.^-  -f  49  =  0,  an  ellipse; 

(3)  3 ^2^  ioxy-\-Ax-—%y=Q,  an  ellipse ; 


^/j:  '9^, 


106  ANALYTIC    GEOMETRY. 

(4:)  ?/^4-  2  a;?/  +  x^—  7  +  1  =  0,  a  parabola  ; 

(5)  y-—  1  +  3 .T  =  (.V  —  ?/)-,  CO)  hyperbola; 

(6)  ?/-=  4  (a;  —  1 ) ,  a  j^arabola; 

(7)  4:a;/y  —16  =  0,  a?t  hyi^erbola. 

81.    T/ie  equation 

Ay'  +  Cx^+Dy  +E^F=  0 
represents  all  species  of  the  conic  sections. 
The  general  equation  of  the  conies  is 

Ay-+Bxy  +  Cx'+Dy  +Ex+F=  0. 

Passing  to  any  rectangular  axes  with  the  sarae  origin  by  the 
formulae  (Art.  22,  Eq.  (8)), 

X  —  .i'l  cos  y  —  v/i  sin  y,         y  =  x^  sin  y  +  2/i  cos  y, 

we  have,  after  omitting  the  subscripts, 

A  {x^  sin^y  +  2xy  siny  cosy  +  ?/  cos^y) 

+ jB  (a^  cos  y  sin y  +  xy  cos'  y  —  xy  sin-  y  —  ?/-  sin  y  cos y) 

+  C  (x?  cos^y  —  2 xy  cosy  sin  y  +  y-  sin-y) 

+  other  terms  not  involving  xy. 

The  term  containing  oyy  is 

[  2 A  sin  y  cos  y  +  -B  (cos^  y  —  sin-  y )  —  2  C  sin  y  cos  y]  xy, 
or         '  [(yl— C)  sin2y -f  iJcos2y]a;y, 

which  will  be  zero  if 

{A-C)  sin2y4-Bcos2y  =  0, 

or  if  tan2y  =  -^I-^;  (1) 

Now  tan2y  can  have  any  and  every  value  from  4-^  to  —  x, 
hence  a  value  can  always  be  found  for  y  which  will  satisfy  (1) 
whatever  the  values  of  A,  7?,  and  C\  that  is,  whatever  the 
species  of  the  conic.     To  (iud  this  value  of  y,  we  have  (Art.  79) 


GENERAL    EQUATIONS   OF   CONIC    SECTIONS.  lUT 

B 

tail  2  Y  = = -. — 7^  = 

F 
2e-  sinu  eosa 


o       ,  7, —  =  tan  2  a, 

1  —  e-  siu-a  —  (1  —  e-  cos-u) 

and  since  tau2a  =  tan  (180°+ 2a),  (1)  will  be  satisfied  when- 
ever 2y=2a  or  180"+ 2a;  that  is,  wheny  =  a  or  90°+ a,  or 
whenever  the  axis  of  the  conic  is  parallel  to  either  axis  of  refer- 
ence.    Hence  every  equation  of  the  form 

Ay-  +  Cx-^  By  +Ex  +  F  =  () 

is  the  equation  of  a  conic  ivhose  axis  is  parallel  to  one  of  the  axes 
of  reference,  and,  since,  B-~AAC  =  —  4:AC  when  B=0,  the 
conic  will  be  an  ellipse,  hyperbola,  or  j^rabola,  according  as 
A  and  C  have  like  signs,  unlike  signs,  or  either  is  zero  (Art.  80) . 
Thus,  whatever  the  signs  or  values  of  D,  E,  and  F, 

Ay-+Cx-+Dy-\-Ex-\-F=0  (2) 

represents  an  ellipse  whose  axes  are  parallel  to  the  axes  of 
reference ;  ^^o_  Q^2_^j)y  _^^j.  +^^  q  (3) 

represents  an  hyperbola  whose  axes  are  parallel  to  the  axes  of 
reference  ;  ^^o _^  j)y  _^  ^^  +  F={), 

or  Cx^  -\-Dy  +  Ex  +  F=0. 

represents  a  parabola  whose  axis  is  parallel  to  X,  or  Y, 
respectively. 

Cor.  rN^'hen  referred  to  the  new  axes  the  coefficients  of 
the  square  are 

A  (cos^y  +  siu'-y)  =A,  C (cos-y  +  sin-y)  =  O, 

or  the  coefficients  of  x'  cindr~<y^are  not  changed  by  this  trans- 
formation. ^^"^ 

Cor.  \^  In  the  circle  5  =  0,  and  A=C  (Art.  49).  Hence 
tan  2  y  =  ^,  or  there  is  always  a  pair  of  axes  parallel  to  the  axes 
of  reference. 


,  .  ,    ^..  ,    ^-.  .    ^      .    i  (^) 


108  A^TALYTIC    GEO.METllY. 

82.  Defs.  The  centre  of  a  circle  is  m  point  equally  distant 
from  every  point  of  the  circle.  Tlie  point  which  has  been 
designated  the  centre  of  the  ellipse,  and  hyperbola,  is  not 
equally  distant  from  every  point  of  these  loci,  but  it  possesses 
a  property  in  common  with  the  centre  of  the  circle,  and  in  virtue 
of  this  common  property  we  may  define  a  centre  for  all  three  of 
these  loci.  A  locus  is  said  to  have  a  centre  lolien  there  is  a  'point 
through  ivhich  if  any  chord  of  the  locus  be  drawn  the  chord  is 
bisected  at  that  point. 

Any  chord  through  the  centre  is  called  a  diameter. 

83.  Every  locus  ivhose  equation  is  of  the  form 

Ay-+Bxy  +  Cx^+  F=Q  ( 1 ) 

has  a  centre. 

For  if  (1)  be  satisfied  for  any  values  x',  y\  of  the  variables, 
it  is  also  satisfied  for  the  values  —  .«',  —y'.  But  the  equation 
of  the  chord  through  {x\  y')  and  {- x\  -y')  is  x'y  —  y'x  =  0 
(Art.  32),  which  passes  through  the  origin  since  it  has  no 
absolute  term.  Moreover,  the  segments  of  the  chord  on  either 
side  of  the  origin  are  equal,  since  the  length  of  each  is  ^x''+y'^. 
Hence  the  locus  has  a  centre,  and  the  centre  is  the  origin. 

Cou.  1.    Every  locus  whose  equation  is  of  the  form 

Ay-  +  av-+F=0 
has  a  centre,  at  the  origin. 

Cor.  2.    The  circle,  ellipse,  and  hyperbola  have  centres. 

84.  The  equation 

Af-+Cx'+F=0 

represents  all  ellipses  and  hyperbolas. 

Resuming  the  general  equation  of  the  conies, 

Ay'+  Bxy  +  Cx-+  lJy  +  Ex  +  F=0,  ( 1 ) 

pass  to  parallel  axes,  the  formuhu  for  transformation  being 

a;  =  av,  +  .i-,,  y  =  y,,-\ry\\ 


GENERAL   EQUATIONS   OF   CONIC    SECTIONS.  109 

and,  after  omitting  subscripts,  we  obtain 

+  Z)(,Vo  +  !/)  +  Eix,  +  a-)  +  F=0.  J 

The  terms  containing  x  and  y  are 

{2  At/,  +  Bx,  +  D)y,   (2  Cx^  +  iJ^o  +  E)x, 

which  will  vanish  if  2Ay,^-{-BxQ-irD=^0,  and  2  Ca;o+-S.yo+£'  =  0  ; 

that  is,  solving  these  equations  for  Xq  and  ?/o,  if  the  new  origin  is 
taken  at  the  point 

,  ^2AE-BD        ^2  CD -BE 
^'       B--iAC  '   ^"       B'-4:AC  ' 

which  is  alwaj's  possible  when  B'—^AC  is  not  zero;  that  is, 
when  the  locus  is  not  a  parabola,  in  which  case  iCo  and  y^  would 
be  iufinit}'.  Hence  the  terms  containing  x  and  y  may  always 
be  made  to  vanish  if  the  locus  is  an  ellipse  or  an  hyperbola,  and, 
when  referred  to  the  new  axes,  the  equation  will  assume  the 
form  Ay'^-{-  Bxy  +  Cx--^  F=  0,  from  which  we  see  that  the  netv 
origin  is  the  centre  (Art.  83).  By  a  second  transformation 
(Art.  81),  the  equation  will  finally  take  the  form 

Ay'^-\-Cc(?^F=0, 

the  central  equation  of  the  ellipse  or  hyperbola  according  as  A 
and  C  have  like  or  unlike  signs. 

Cor.  1.  Since,  when  B'—  AAC=  0,  a'o  and  yo  are  infinity,  the 
parabola  has  no  centre. 

CoR.  2.  Since  the  above  values  of  .r,,  and  ?/o  are  independent 
of  F,  central  conies  ivhose  equations  differ  only  in  their  absolute 
terms  are  concentric. 

Cor.  3.  By  examining  Eq.  (2)  we  see  that  the  first  three 
terms  of  the  equation  are  not  altered  by  the  transformation. 

85.   Varieties  of  the  parabola. 

We  have  seen  that  when  B'—  4:AC=  0  the  centre  is  at  in- 
finity, and  that  therefore  the  terms  Dy  and  Ex  cannot  be  made 


110  ANALYTIC   GEOMETKY. 

to  vanish  from  the  general  equation  when  it  represents  a  parab- 
ola; also  (Art.  81)  that  the  term  Bxy  will  vanish  if  either  axis 
of  reference  is  assumed  parallel  to  the  axis  of  the  parabola,  in 
which  case  -B'—  4^1C  becomes  —  A  AC,  and  eilher  A  or  C  must 
be  zero.     Making  then  B  =  0  and  C  =  0  in  the  general  equa- 

*'°° '  Af  +Dy-[-  E.V  -j-F=0  ( 1 ) 

represents  all  parabolas.  To  see  if  this  form  can  be  still  fur- 
ther simplified,  transform  to  new  parallel  axes  by  the  formulae 
X  =  Xo-i-Xi,  y  =  ?/„  +  llu  ^ud  we  have,  omitting  subscripts, 

Af+{I)  +  2Ay,)y  +  Ex  +  Ay,'+  I)y,  +  Ex,  +  F=0. 

As  the  terms  containing  x  and  y  cannot  both  be  made  to  vanish, 
let  us  see  if  one  of  them,  as  ?/,  and  the  absolute  term  can  be 
made  to  vanish.     This  requires  that 

D  +  2  Ay,  =  0  and  Ay^  +  By,  +  Ex,  +  F  =  0, 

or  that     ?/o  =  —  — -  and  a^o  =  — — -— — 
2  A  4  AE 

The  equation  then  assumes  the  form  Ay- -\-  Ex  =  0,  or 

^  A    ' 

which  is  the  equation  of  the  parabola  referred  to  its  vertex  and 
axis  (Art.  74),  the  curve  lying  to  the  right  or  the  left  of  the 
origin  according  as  E  and  A  have  unlike  or  like  signs.  Hence 
the  disappearance  of  the  absolute  term  and  that  containing  y 
involves  a  svstem  of  reference  toliose  origin  is  the  vertex.  This 
transformation  is  always  possible,  except  in  two  cases :  First, 
when  E  —  0,  in  which  case  x,  =  oo.     Equation  (1)  then  becomes 

Ay^  +  Dy  +  F  =  0 ,  or 

-D±Viy^^AAF 

'  = 2A ' 

which  represents  two  straight  lines  parallel  to  X,  real  and  dif- 
ferent, real  and  coincident,  or  both  imaginary,  according  as  D^ 
is  greater  than,  equal  to,  or  less  than  4:AF.  These  are  the 
particular  cases  of  the  parabola,  the  vertex  receding  to  infinity. 


GENERAL   EQUATIONS   OF   CONIC    SECTIONS.  Ill 

/Second,    when  A  =  0,  iu    which   case,  however,  the   equation 
ceases  to  be  one  of  the  second  degree. 

86.  Defs.  A  diameter  of  a  conic  has  been  defined  as  a  chord 
through  the  centre.  As  the  parabola  has  no  centre  it  would 
appear  that  it  has  no  diameters.  A  set  of  lines  may,  however, 
be  drawn  to  the  parabola  which  possess  a  property  in  common 
with  the  diameters  of  the  ellipse  and  hyperbola  ;  and  in  virtue 
of  this  common  property  we  may  define  a  diameter  for  all  three 
species  of  the  conies. 

A  diameter  of  a  conic  is  the  locus  of  the  middle  points  of  par- 
allel chords. 

87.  To  Jind  the  locus  of  the  middle  points  of  parallel  chords. 
First.    For  the  ellipse  and  hyperbola. 

Let  Ai/+O:r+F=0,  (1) 

in  which  A  =  a-,  C  =  ±  Ir,  J^=  qp  «"^"^  ^s  the  conic  is  an  ellipse 
or  an  hyperbola,  be  the  equation  of  the  locus,  and 

y  =  a'x  +  b'  (2) 

that  of  an}^  chord  PQ.     Combining  (1)  and  (2)  to  find  the  inter-, 
sections  P  and  Q,  we  have,  after  substituting  y^  from  (2)  in  (1), 

^        2a'bA  ^^      b"A  +  F 
a"A  +  C^  a"A  +  C 


Fig.  54. 


112 


ANALYTIC    GEOMETRY. 


or,  represeuting  the  coefficient  of  x  by  q  and  the  absolute  terra 

by  r, 

xr  -f-  qx  =  ?•, 


whence 


Fig.  55. 


which  are  the  abscissas  of  P  and  Q.     Substituting  these  values 
of  X  in  (2),  we  find  the  ordinates  of  P  and  Q  are 


y  =  a 


,(-i±*L.  ,  ^ 


\r  +  l    +b' 


Now  the  coordinates  of  the  middle  poifit  31  of  PQ  are  given 
by  the  formulae  x  =  '- — - — i   y  =  ' — ^-^—      Taking,   therefore, 

the  half-sum  of  the  above  values  of  x  and  y,  we  have  for  the 

coordinates  of  3f  ,,i^ 

x=-l    y=--^+b', 

or,  replacing  the  value  of  q, 

a'b'A 


x=  — 


^  _    a"b'A        ,, 


GENERAL   EQUATIONS    OF   CONIC    SECTIONS. 


113 


For  all  other  chords ^>tuaWe/  to  PQ,  a'  remains  the  same,  but 
h'  differs.  Eliminating  then  b'  by  substituting  its  value  from 
the  first  in  the  second  of  the  above  equations,  we  obtain 


y  = 


c 

a'A 


x=^. 


b- 
a'a? 


X, 


(3) 


which  is  a  relation  between  the  coordinates  of  the  middle  points 
of  all  chords  parallel  to  PQ ;  it  is  therefore  the  equation  of  a 
line  through  these  middle  points.  Being  of  the  first  degree  it  is 
a  straight  line,  and  having  no  absolute  term  it  passes  through 
the  origin,  which  is  the  centre.  Hence,  the  locus  of  the  middle 
points  of  parallel  chords  to  the  ellipse,  or  hyperbola,  is  a  straight 
line  throngh  the  centre. 

CoK.    If  A  =  C,  or  the  locus  is  a  circle,  (3)  becomes 

1 

2/=  --^^ 
a' 

which  is  perpendicular  to  y  =  a'x  +  b'. 
Second.    The  p)arabola. 

Let  2/"  =  2jXT  be  the  parabola,  and  y  =  a'x  Jf-  b'  any  chord  PQ. 

Combining  as  before, 

2a'b'-2p    _      ly^ 


a'' 


a" 


or,  x"^  +  qx  =  r,  whence,  in  the  same  manner  the  coordinates  of 
the  middle  point  M  are 


x  = 


9 
2' 


u'q      , , 


or,  replacing  q  by  its  value, 

a'b'  —  »  p 

""  a'''  a' 

from  which  we  see  that  the  abscissa  x  of 

the  middle  point  varies  with  b\  but  that 

the  ordinate  y  is  constant  if  a'  is  constant ; 

that  is,  if  the  chords  vlxq  ptarallel.    Hence,  the  loctis  of  the  middle 

points  of  parallel  chords  to  the  parabola  is  a  straight  line  parcdlel 

to  X. 


Fig.  56. 


114  ANALYTIC   GEOMETRY. 

The  studeut  will  observe  that  if  a  diameter  be  defiued  as  a 
chord  through  the  centre,  the  diameters  of  the  parabola  are 
necessarily  parallel  as  the  centre  is  infiniteh-  distant. 

The  extremities  of  any  diameter  are  called  its  vertices. 

88.  The  tangents  at  the  vertices  of  a  diameter  are  parallel  to 
the  chords  bisected  by  that  diameter. 

Since  the  diameter  TT  (Figs.  54,  55,  56)  bisects  all  chords 
parallel  to  PQ,  as  J/ approaches  T  (or  T'),  P  and  Q  approach 
each  other,  and  J/P,  J/Q,  remaining  equal,  must  vanish  together. 
Hence,  when  31  coincides  with  T  (or  T'),  PQ  will  have  but 
one  point  in  common  with  the  curve,  or  is  a  tangent. 

89.  Def.     One  diameter  is  said  to  be  conjugate  to  another 
hen  it  is  parallel  to  the  tangents  at  the  vertices  of  the  latter. 


ic 


90.    Conjugate  diameters  of  the  ellipse. 

Let  KK'  (Fig.  54)  be  drawn  parallel  to  the  tangent  at   T, 
that   is,  parallel  to  PQ.     Its  equation  will  he  y  =  a'x.     The 

equation   of    TT'   is   y=  a"x=-—  x    (Art.    87,    Eq.    3). 

7  2  «  "" 

Ilcnce  o'a"  = ^  is  the  relation  which  must  exist  between  the 

a- 

slopes  of  a  diameter  and  the  chords  which  it  bisects.  But  this 
relation  is  satisfied  for  KK'  and  the  chords  PQ',  etc.,  parallel 
to  TT.  Hence,  if  one  diameter  is  conjugate  to  another,  the 
latter  is  conjugate  to  the  former,  and  the  tangents  at  the  vertices 
of  conjugate  diameters  form  a  ^xirallelogram. 

a'a"=-^l  (1) 

a^ 

is  called  the  equation  of  condition  for  conjugate  diameters  to  the 
ellipse.  Since  the  rectangle  of  their  slopes  is  negative,  the 
tangents  of  the  angles  which  they  make  with  X  have  opposite 
signs ;  hence,  if  one  diameter  makes  an  acute  angle  with  the 
transverse  axis,  the  other  will  make  an  obtuse  angle,  or  conju- 
gate diameters  to  the  ellipse  lie  on  opposite  sides  of  the  conjugate 
axis. 


GENERAL   EQUATIONS    OF   CONIC    SECTIONS. 


115 


Cor.    If  a  =  b,  (1)  becomes  a'= ,  or  conjugate   diam- 

a" 

eters  to  the  circle  are  at  right  angles  to  each  other. 

91.  Every  straight  line  through  the  centre  of  an  hyperhola^ 
except  the  diagonals  of  the  parallelogram  on  the  axes,  meets  the 
hyperbola  or  the  conjugate  hyperbola. 

Let  y=  a'x  (1) 

be  any  straight  line  through  the  centre, 

ay  -  Vx"  =  -  a'b^  (2) 

the  equation  of  the  X-hyperbola,  and  (Art.  71) 

ay  _  6V  =  a-b'  (3) 

that  of  the  I"-hyperbola.     Combining  (1)  in  succession  with  (2) 
and  (3),  we  have 

a-b' 


o  _       a-b~ 


b'  -  d'a 


/-V,'2 


(4) 


XT  — 


a'a"  -  b^ 


(5) 


Now  if  a'  <,-■,  X  is  real  in  (4)  and  imaginary  in  (5),  and  the 


Fig.  57. 


116 


ANALYTIC    GEOMETllY. 


line  intersects  the  X-hyperbola,  as  TT'.     If  a'  >  —,  x  is  imag> 

(Jj 

inary  iu  (4)  and  real  in  (5),  and  the  line  intersects  the  F-hyper- 
bola,  as    KK'.     If  a'  =  ±    ,  both  values  of  x  are  infinity.     In 

this  case  (1)  becomes  y—±  -.r,  the  equations  of  CS  and  CS', 

a 

the  diagonals  of  the  rectangle  on  the  axes,  neither  of  which 
meet  either  hyperbola  within  a  finite  distance. 

92.    Defs.     The  diagonals  of  the  rectangle  on  the  axes  of  a 
pair  of  conjugate  hyperbolas  are  called  the  asymptotes.     Their 

equations  being  y  =±-x,  if  a  —  h  their  included  angle  is  90°, 

a 

and  the  hyperbola  is  said  to  be  rectangular  ;  or,  n^hen  an  hyper- 
bola is  rectangular  it  is  also  equilateral  (Art.  71). 


93.   Conjugate  diameters  of  the  hyperbola. 

Of  ttvo  conjugate  diameters,  one  meets  the  X-,  the  other  the 
Y-hyperbola. 

Let   TT'  be  any   diameter   bisecting  a  system   of    parallel 


Ki),'.  .-18. 


GENERAL   EQUATIONS    OF   CONIC    SECTIONS.  117 

chords  of  which  PQ  is  oue.  Draw  KK'  parallel  to  PQ^  that 
is,  to  the  tangents  at  the  vertices  of  TT' ;  it  is  then  conjugate 
to   TT'.     Being   parallel    to  PQ,  its  equation   is  ?/  =  a'.t',  and 

that   of    TT'  is   y  =  a"x  =  -^x    (Eq.   3,   Art.   87).       Hence 

a'a"  =  —  is  the  relation  which  must  exist  between  the  slopes  of 
a-  , 

a  diameter  and  the  chords  which  it  bisects.     If  a'  <    ,  a"  must 

7  7  2  « 

evidently  be  >  -,  since  their  product  =  — ,  and  conversely  ;  or, 

h "  ^''  '^''" 

since  -  is  the  slope  of  the  asymptote,  if  one  diameter'  intersects 
a 

the  X-liyperhola,  its  conjugate  will  intersect  the  Y-hyperhola,  and 
conversely. 

Again ;    since  the  equation  of   the    "1^- hyperbola   is  derived 
from  that  of  the  X-hyperbola  by  changing  the  signs  of  a^  and 

If  (Art.   71),  a'(7"  =  —   is  also  the  relation  which  must  exist 

cr 

between  the  slopes  of  any  diameter  of  the  I'-hyperbola  and  the 
chords  which  it  bisects.  But  this  relation  is  satisfied  for  KK' 
and  the  chords  P'Q',  etc.,  parallel  to  TT' \  hence  TT' is 
parallel  to  the  tangents  at  K  and  K' ,  or  is  conjugate  to  KK' ; 
hence,  if  one  diameter  is  conjugate  to  another,  the  latter  is  conju- 
gate to  the  former,  and,  as  in  the  case  of  the  ellipse,  the  tangents 
at  the  vertices  of  conjugate  cjiaineters  form  a  jKirallelogram. 

The  equation  a'a"  —  — 

a- 

is  called  the  equation  of  condition  for  conjugate  diameter's  to  the 
hyperbola.  Since  a'a"  is  positive,  the  angles  which  two  conju- 
gate diameters  to  an  hyperbola  make  with  the  transverse  axis 
are  both  acute,  or  both  obtuse,  or  the  diameters  lie  on  the  same 
side  of  the  conjugate  axis. 


CONSTRUCTION  OF  CONICS  FROM  THEIR  EQUATIONS. 

94.    First  jMethod.     By  comparison  ivith  the  general  equa- 
tion. 

Make   the   coefficients  of   like  terms  in  the  given   and    the 


118 


ANALYTIC    GEOMETRY. 


general  equation  equal  by  dividing  each  equation  by  the  co- 
efficient of  the  same  term.  Equating  the  resulting  coefficients 
of  corresponding  terms,  we  have  five  equations  from  which 
a,  m,  n,  e,  and  p,  may  be  determined.  This  method  is  tedious 
and  of  little  practical  value  except  as  e  =  1,  or  some  of  the  co- 
efficients are  zero. 

Example.  1.  ?/2  +  4?/  +  4.i;  +  4  =  0.  Since  B'  —  4 AC  =  0, 
the  conic  is  a  parabola,  and  therefore  e=  1.  The  coefficient  of 
y-  being  unity,  divide  the  general  equation  (Eq.  1,  Art.  79)  by 
the  coefficient  of  i/,  1— e^sin^a,  and  we  have,  after  making 
e=l, 


V 

\ 

P 

-^ 

0 

Q 

i  ^ 

F    j 

^ 

s 

D' 

Fig.  59. 


—  2  sin  a  cos  a 
1  —  sin-u 

=  0, 

(1) 

1  —  COS"  a 

1  —  sin- a 

=  0, 

(2) 

2j:»sintt  —  2n 

—  4 

('^) 

1  —  sin- a 

^1 

2/?  cos  a—  2  m 
1  «r-sin-a 

=  4, 

(4) 

m-  +  11-  —  p- 
1        ■  ■> 

=  4. 

(^) 

1 


sm-a 


From  (1),  -2sinacosa=0;  .-.a  must  be  0°,  90°,  180°,  or 
270°.  From  (2),  cosa  =  ±  1  ;  hence  a  cannot  be  90°  or  270°, 
and  is  either  0°  or  18U°.  In  either  case  (3)  gives  n  =  —  2. 
Substituting  cosa  =  ±l  in  (4),  we  have  ±  2])  —  2m  =  4,  or 
111  =  ±p  —  2,  according  as  a  is  0°  or  180°.  From  (;')),  since 
7i  =  — 2,  m'^  =  p'^\  or,  substituting  the  above  values  of  w, 
j)  =  ±l.  Bnt\2^  is  always  positive;  taking,  therefore,  the 
upper  sign,  a  =  0°.  Finally,  from  (4).  making  cosa=l  and 
p=l,  we  have  m  =  —\.  Tlie  values  of  the  constants  are 
thus  :  r'  =  1 ,  a  =  0°,  m  =  —  1 ,  ?«  =  —  2,  j)  =  1 .  To  construct 
these  results,  lay  off  OQ  =p=\  to  the  right,  since  a  =  0°,  and 
draw  the  directrix  Z)7)'  perpendicular  to  X.  Construct  F^ 
(  —  1,   —2),  and   through  F  draw  FS  perpendicular   to  DD'. 


GENEKAL    E(^)UAT10N8    OF    CONIC    SECTIONS.  119 

Having  thus  the  focus  aud  cHrectrix,  the  parabola  may  be  cou- 
sti'ucted  as  in  Art.  TG. 

95.    Second  Method.     Bii  transformation  of  axes. 

If  B-  —  4:AC  is  not  zero,  the  conic  is  an  ellipse  or  hyperbola, 

imd 

.  ^  2AE-BD     ^    ^  2  CD -BE 

are  the  coordinates  of  the  centre  (Art.  6-i).  Transferring  to 
parallel  axes  with  (a^o,  y^)  as  a  new  origin,  we  have  the  equation 
of  the  ellipse,  or  hyperbola,  referred  to  its  centre.  If  the  term 
Bxy  is  not  present  in  the  primitive  equation,  the  result  of  this 
transformation  is  the  central  equation  of  the  ellipse,  or  hyperbola. 
If,  however,  this  term  is  present,  we  must  transfer  again  to  new 
axes  with  the  same  origin,  the  angle  between  the  new  and  primitive 

T> 

axes  of  X  beiuo-  determined  by  the  condition  tau2Y  = 

^  ^      A-C 

(Art.  81).  If  B'  —  ■XAC=  0,  the  conic  is  a  parabola.  Trans- 
fer first  to  new  axes  with  the  same  origin,  the  new  axes  of  X 

being  subject  to  the  condition  tan2y  = — ;  then  to  parallel 

axes  M'hose  origin  is  (Art.  8o) 

_  I^-±AF        _-D 

'"'-   4AE  '  '^"-yz' 

the  resulting  equation  will  be  the  equation  of  the  parabola 
referred  to  its  vertex  and  axis. 

Examples.     1 .  5  if-  -\-2xy  -{-  ox-  —  12 y  —  V2x=0. 

B'-4AC=-96, 

hence  the  conic  is  an  ellipse  and  has  a  centre.  The  coordinates 
of  the  centre  are 

^^2AE-BD^^  ^2CD-BE^ 

'      B'-4:AC        '     "^^       B--4AC 

and  the  fornmhTe  of  transformation  are 

X  =  .(•„  +  X,  =  1  +  a'l,     y  =  y,  +  y,  =  1  +  ?/,. 


120 


ANALYTIC    GEOMETRY. 


-4. 


Substituting  these  in  the  given  equation,  omitting  subscripts, 

we  have  .   ,  ,    ^        ,   c    -^      .  ^      r> 

o/  4-  2 x'^  +  5 .1"  —  12  =  0. 

To  obtain  the  central  equation  we  must  have 

-B 


tan  2  y  = 


•  =  —  GO 


y 


=  -  4.7 


A-C 
and  the  formulae  of  transformation  are 

X  =  .Ci  cos  7  -  ?/i  sin  y  =  V^  {x^  +  y^ , 

y  =  .Ti  sin  y  +  2/1  cos  y  =  Vi  (2/1  -  x^ ) . 

Substituting    these   values    in    5?/- +  2a;?/4-5x'- —  12  =  0,    and 

omitting   subscripts,  we  obtain    3?/^ -f  2x-^=  6.     The  axes  are 

therefore    2  V3    and    2  V2, 


and     the     eccentricit}^ 


1 

V3* 

To  construct  the  ellipse, 
construct  ( 1 ,  1 ) ,  the  new 
origin  Oi,  and  draw  OjXi, 
Oi  ^'i,  the  parallel  axes. 
Draw  OiX,  making  the 
angle  Xi  Oi  X2  =  —  45°,  and 
OxY.,  perpendicular  to  it. 
On  O1X2  lay  off 

0i.l=0i.4'=  V3, 

and  on  OxY^i  OiB=  0^0=  V2.  AA'  and  OjB  are  the  axes  of 
the  ellipse  ;  the  focus  may  be  found  as  in  Art.  58,  and  the 
ellipse  constructed  as  in  Arts.  54  and  60.  The  curve  may  be 
traced  with  approximate  accuracy  by  determining  the  intercepts 
on  the  axes.  Thus,  from  5y^  + 2a;i/ 4-5ar  —  12y  —  12.i;  =  0, 
x  —  Q  gives  ?/  =  0  and  ^-  ( 0  and  ^)  ;  .v  =  0  gives  x  =  0  and 
^  {0  and  9) •  In  the  same  way  from  5 y'  +  2xy  +oxr  —  12  =  0, 
0,S=OiT=  V/,  OiU=  OiF=  V-U.  Tln-ough  the  points 
thus  found  trace  the  curve. 


Fig.  60. 


2.  y^-2xy  +  x^  +  8x-lC>  =  0 
conic  is  a  parabola.     tan2y  = 


jr--iAC=0,  hence   the 


B 


A-C 


.—  X, 


y  =  45°,  and  the 


GENERAL   EQUATIONS   OF    CONIC    SECTIONS. 


121 


^0^ 


2A 


V2. 


formulae  of  Iransformation  are  x  =  V^  {x  —  ?/) ,  y  =  Vi  {x  +  y) , 
tiud  the  transformed  equation  2y-  —  4  V2  2/  +  4  V2  .i-  —  16  =0. 

From    the    latter,    .r,,  = ^^ —  = , 

Transferring  to  parallel  axes 

with    the   orio;in  (  ,  V2  ), 

VV2  / 

we  find  ?/-  =  —  2  V2  x.  To 
construct  the  parabola,  draw 
the  axes  Y^OX^^  making 
XOXi  =  45°.  On  these  axes 
construct  the  vertex 


,  V2  ],  or  Oi, 


Fig.  61. 


V2 
and  draw  the  parallel  axes 
Xo  Oi  Y^.  We  may  now  con- 
struct the  parabola  whose 
parameter  is  2  V2  as  in  Arts.  74  or  76,  or  determine  the  inter- 
cepts and  trace  the  curve  approximatively.  OQ  =  —  4  +  4  V2, 
OQ'=-4-4V2,  OR  =  OR'  =  i,  0*S=2V2,  Or  =  V2  + ViO, 
0T'=  V2- VIO. 

3.  y~  —  x;-  +  y  —  X  +  2  =  0.  B-  —  4AC  =  4,  hence  the  conic  is 
an  hyperbola,  its  axes  being  parallel  to  the  axes  of  reference, 
since   B  =  0. 


•2  ' 


yo  =  —  i-  Passing  to 
parallel  axes  whose  ori- 
gin is  (-1,  — I),  we 
have  y~  —  a;-  =  —  2,  an 
equilateral  hyperbola 
whose  axes  are  2  a/2, 
eccentricit}'  is  V2,  and 
cutting  the  primitive 
axis  of  X  at  1  (Q)  ^^^ 


Y' 


O' 


0 


Fig.  62. 


122 


ANALYTIC   GEOMETRY. 


4.  ?/2  -f  2  V.'5  .rji  -  X-  -  64  =  ().      B'  —  4:AC=\^,  hence   the 

conic  is  an  hyperbola  referred 
to  its  centre  ( Art.  83 ) . 

tan2y=-V3,  .•.2y=-(;0^ 

or  y  =  -30°.  Transferring 
to  new  axes  such  that 

-^  XOX,  =  -  30°, 

the  equation  becomes 

i-/2_x2=32, 

the  equilateral  F-hyperbola 
whose  axes  are  8  V2,  cutting 
the  primitive  axis  of  I"  at 
±8  (Qand  Q'). 

5.  y"  -  Axy  +  fx^  +  2?/  -  2a-  +  3  =  0. 

6.  |ar  +  4.T?/  +  3r-3  =  0. 

96.    Third  Method.     By  conjugate  diameters. 
The  general  equation  of  a  conic  being 

Ay--\-  Bxy  +  Cx''+  Dy  -^Ex  +  F^O, 
solving  for  y,  we  have 


0      Bx  +  D            Cx-+Ex  +  F 
y  "1  -, — .'/  — : » 


A 


whence 

Bx+D 


y=- 


1 


■lA 


±^^-^{B'-4.AC)x;'+2{BD--2AE)x+D'-4.AF. 


First.    Construct  the  line  QR  whose  equation  is  y— 


Bx+I) 

2A 


Every  value  of  x  locating  a  point  M  on  this  line  locates  two 
points  P'  and  P"  of  the  locus,  equally  distant  from  Qli  and  on 
opposite  sides  of  it,  this  distance  being  the  radical  in  the  value 
of  ?/.  Hence  QR  bisects  a  system  of  chords  parallel  to  Y  and 
is  a  diameter. 


GENERAL   EQUATIONS    OF   CONIC    SECTIONS. 


123 


Second.  Values  of  x  which  reuder  the  radical  zero  give  the 
same  values  for  y  for  both  the  locus  and  the  diameter ;  hence 
the  values  of  x  found  from  the  equation 

(B--  4:  AC)  x--\-  2  (BD  -  2  AE)  x  +  D'-AAF=  0 

determine  the  points  where  the    conic    cuts  QR ;    that  is,  the 


Fig.  64. 


vertices  T,  T',  of  the  diameter.  This  equation  being  a  quadratic, 
there  will  be  two  such  points  except  when  B'-  —  4:AC=  0,  in 
which  case  the  conic  is  a  parabola  and  there  will  be  but  one 
vertex.  If  the  conic  is  an  ellipse  it  lies  wholly  between  Tand  T" ; 
if  an  hyperbola,  w4iolly  without  these  points.  In  either  case 
the  half  sum  of  the  above  values  of  x  determines  the  centre  C, 
and  the  corresponding  values  of  //  locate  K  and  K',  the  vertices 
of  the  conjugate  diameter.  Having  thus  the  circumscribing 
parallelogram  (Arts.  90,  93),  a  few  other  points  may  be  con- 
structed, especially  the  intercepts  on  the  axes,  and  the  curve 
sketched  with  sufficient  accuracy. 


Examples.   1 .  4 ?/-  +  4 x>/  -{-5xr  —  8y  —  28 x  +  24 
B--^AC=-64, 
and  the  conic  is  an  ellipse.     Solving  for  y  we  find 


=  0. 


y  =  ^  (2  -  x)  ±  V-  X-+  6x-o. 


124 


ANALYTIC    GEOMETUY. 


Construct  the  diameter 

y  =  ^(2-x),  QB. 
Placing  —  0^  +  6a;  —  5  =  0, 

we  find  x=  5  and  1,  whence 

7/  =i  (2  —  if)  =  —  f  and  ^.  or 
(5,— f)  and  (1,  i)   are  the  vertices  T  and   T.     The  abscissa 


Fig.  65. 


of  C  is  ^  (5  -f- 1)  =  3,  whence,  from  the  equation  of  the  conic 
2/ =  4  and  —  f,  locating  K  and  7i'.  The  circumscribing  paral- 
lelogram may  now  be  drawn.     Making  ?/  =  0  we  find  the  X- 

'  —      Intermediate   points   uui}^  be  found 


intercepts  = 


o 


if  necessary  ;  thus,  for  x=A,  y  =  —l  ±  V3,  locating  P'  and  P". 
Trace  the  curve  through  the  points  tlius  found,  tangent  to  the 
circumscribing  parallelogram  at  K,  K',  T  and  T'. 

2.  2/2  +  2a;2/  +  ar'  +  2.v-7.c-8  =  0. 


GENERAL  EQUATIONS   OF   CONIC   SECTIONS. 


125 


B'  —  4AC=  0,  and  the  conic  is  a  parabola.     Solving  for  y, 

tj  =  -  (.f+l)±  V9a'+9.  Y 

Construct  the  diameter  QR, 

y  =  -{x  +  \). 

The  radical  gives  but  one  value 
of  x  =  —  l,  for  which  y  =  0,  lo- 
cating tlie  vertex  T.  The  X- 
intercepts  are  8,  —  1,  and  the 
F-intercepts  2,  —4.  Interme- 
diate points  may  also  be  found  ; 
thus  for  x  =  S,  y  —  2  and  — 10 
(P'  and  P").  Trace  the  curve 
through  these  points  and  tangent  at  T  to  a  parallel  to  Y 

3.    y^  +  2xy-2x:^-4:y-x+l0=0. 
B--4.AC=  12, 
.-.  the  conic  is  an  hj'perbola. 


Fisr.  66. 


y  =  -  {x  -  2)  ±  Vdx--3x-G. 

The  diameter  is  y  =  —  (x  —  2),  its 
vertices  are  (2,  0)  and  (  —  1,  3). 
The  X-intercepts  are  2,  —4,  the 
F-intercepts  being  imaginary. 

4.  y^  +  2xy-^3x^-4:x  =  0. 

5.  y^  —  2 xy  -\-  X-  —  y  -\-  2  X  —  1  =  0. 

6.  y--2xy-\-x''-\-x=0. 


Fig.  67 


97.  When  the  equation  of  the  conic  does  not  contain  the 
term  involving  xy,  the  axes  of  the  conic  are  parallel  to  the  axes 
of  reference,  and  its  position  may  be  determined  by  the  prin- 
ciples of  Art.  17.  If  the  squares  of  both  variables  are  present, 
it  is  an  ellipse  or  an  hyperbola  according  as  their  signs  are 
like  or  unlike ;  if  these  coefficients  are  equal  in  magnitude 
and  sign,  it  is  a  circle ;  if  numerically  equal  and  of  opposite 


126  ANALYTIC   GEOMETRY. 

signs,  an  equilateral  hyperbola.  Solving  the  equation  for  either 
variable,  as  y,  values  of  x  which  render  the  radical  part  of  y 
zero  give  the  extremities  of  the  axis  parallel  to  X,  and  the 
algebraic  difference  of  these  values  is  the  length  of  this  axis  ; 
the  half  sum  of  these  values  of  x  is  the  abscissa  of  the  centre, 
and  the  corresponding  values  of  y  determine  the  vertices  of  the 
axis  parallel  to  Y^  then*  algebraic  difference  being  its  length. 
If  the  term  containing  x  is  lacking,  the  centre  is  on  Y;  if  the 
term  containing  y  is  absent,  the  centre  is  on  X. 

If  the  equation  involves  the  square  of  but  one  variable,  the 
conic  is  a  parabola  whose  axis  is  parallel  to  the  other  axis  of 
reference,  and  coincides  with  it  when  the  the  first  power  of  the 
variable  whose  square  enters  the  equation  is  lacking.  The 
vertex  is  found  by  solving  the  equation  for  the  variable  which 
enters  as  a  square  and  placing  the  radical  part  equal  to  zero ; 
this  equation  determines  the  limit,  i.e.,  the  vertex. 

Examples.    1.    9y/_|-4.T- -  .36?/ -  8.^•  + 4  =  0.       A    and    C 

have  like  signs,  .*.  the   conic  is  an    ellipse.      Solving    for   ?/, 

2/  =  2  ±  ^  V— 4.X-  +  8X-  +  32.     The  limits  along  X  are   found 

from  —  4a^  +  8a;  +  32  =  0  to  be  4  and  —  2,  and  the  axis  parallel 

4  —  2 
to  X  is  therefore  6.     The  abscissa  of  the  centre  is  ■  =  1, 

2  ' 

and  the  corresponding  values  of  y  are  4,  0,  or  the  axis  parallel 

to  Y'\s  4.     Hence  the  locus  is  an  ellipse  whose  centre  is  (1,  2) 

and  axes  6  and  4,  its  transverse  axis  being  parallel  to  X. 

2.  ?/-  +  4 jy  —  6.^  —  14  =  0.  The  locus  is  a  pai*abola,  its  axis 
being  parallel  to  X.  Solving  for  y,  y  =  —  2  i  VGa;  -j-  18  ; 
hence  its  vertex  is  (—3,  —  2). 

3.  4/  +  .i^+ 16?/-4.^•+lG  =  0. 

4.  92/--4.r2-36y  +  24a;- 36  =  0. 

GENERAL   THEOREMS. 

98.  Tlirough  any  jive  points  in  a  plane  one  conic  may  be 
made  to  jjuss. 


GENERAL   EQUATIONS   OF   CONIC   SECTIONS.  127 

Let  (.fi,  y,)^  (%'  ?/-'),  (-^3,  ys),  (■>',.  /a),  (a-55  .Vs),  be  the  five 
given  points.  Dividing  the  general  equation  of  a  conic  by  the 
coefficient  of  any  of  its  terms,  and  distinguishing  the  new 
coefficients  by  accents,  we  have 

A'f--{-B'xy+C'x''-\-D'y  +  E'x  + 1  =  0.  (1) 

Substituting  in  succession  the  coordinates  of  the  given  points, 
since  the  conic  is  to  pass  through  them,  we  have 

A'yf  +  B'x.y,  +  C'x'^  +  D'y,  +  E'x,  +  1=0,^ 

A '2/2-  +  B'x.y.  -f  C'av  +  D'y,  +  E'x^  +1=0, 

Ahii  +  B%y,  +  C'x^  +  D'y,  +  E'x,  +  1=0,  }  (2) 

A'y,'  +  B%y,  +  C%'  +  D'y,  +  E'x,  +  1  =  0, 

A'y-J'  +  B'.^5?/5  +  C  'av  + 1>'^,  +  E'x,  +  1  =  0, 

in  which  A',  B',  C",  Z>',  and  E',  are  the  only  unknown  quanti- 
ties, and  from  which  their  values  may  be  determined  by  elimi- 
nation. Since  these  equations  are  of  the  first  degree,  each  of 
these  quantities  has  but  one  value.  Substituting  in  (1)  the 
values  of  A',  B',  etc.,  found  from  (2),  the  resulting  equation 
will  be  that  of  the  conic  passing  through  the  five  given  points. 
If  one  of  the  points  is  the  origin,  one  of  the  equations  (2) 
would  be  1  =  0,  which  is  impossible.  In  such  a  case  divide  the 
general  equation  by  any  coefficient  except  the  absolute  term. 
This  results  from  the  fact  that  the  equation  sought  can  have  no 
absolute  term. 

Examples.  1 .  Find  the  equation  of  the  conic  passing  througli 
(4,4),  (4,-4),  (9,6),  (9,-6),  (0,0). 

Since  the  conic  is  to  pass  through  the  origin,  F=  0. 

Dividing  the  general  equation  by  A,  we  liave 

f-  +  B'xi/  +  C'x'-2  +  D'y  +  E'x  =  0. 

Substituting  the  coordinates  of  the  remaining  points, 

16+16i?'+ 16C"  +  4Z''  +  4£'  =  0.  (1) 

16-l6B'+l6C'-iD'  +  'iE'  =  0.  (2) 

36  +  biB'+8\C'+6D'  +  9E'  =  0.  (3) 

36-54jB'  +  81  C"-0Z>'+9£'  =  0.  (4) 


128  ANALYTIC    GEOMETRY. 

From  (1)  and  (2),  and  (3)  and  (4),  by  addition, 

32+   32C'+    8^'  =  0.  (5) 

72  +  162  C"  +ISE'  =  0.  (6) 

Eliminating  E'  between  these  we  find  C"  =  0,  which  in  (5)  gives  E'  =  —  i. 
Substituting  these  values  in  (1)  and  (3),  we  have 

54  B'  +  6  Z>'  =  0, 
whence  B'  =  0,  D'  =  0.     The  required  equation  is  therefore  y~-—4:X. 

2.  Fiud  the  equatiou  of  the  conic  passing  through  (—1,  2), 
(-hi)^  (-1,1),  (-1,  i),  (-4,8). 

Ans.  i/  +  x^  +  2xy  -\-3x  —  y  -{-iz=  0. 

3.  Find  the  equation  of  the  conic  passing  through  (5,  0), 
(0,  5),  (-5,  0),  (0,  -5),  (0,  0). 

In  this  case  F  =  0,  since  one  of  the  points  is  the  origin.  Divide  the 
general  equation  by  B,  otherwise  the  term  Bxi/  will  disappear  for  every 
substitution,  and  B  will  be  undetermined.  Ans.  The  Axes. 

4.  Througli  how  many  points  may  the  conic 

be  made  to  pass? 

5.  Find  the  equation  of  the  circle  circumscribed  about  the 
triangle  whose  vertices  are  (;5,  1),  (2,  3),  (1,  2). 

Ans.  3?/-  + 3a;' -11?/ -13a; +  20  =  0. 

6.  Find  the  equation  of  a  circle  through  the  origin  and  mak- 
ing intercepts  a  and  h  on  the  axes. 

99.    Two  conies  can  intersect  in  hut  four  points. 

The  coordinates  of  the  points  of  intersection  of  two  conies 
will  be  found  by  combiniug  their  equations  (Art.  36).  But  we 
know  from  Algebra  that  the  elimination  of  one  unknown  quan- 
tity from  two  quadratic  equations  gives  rise,  in  general,  to  an 
equation  of  the  fouith  degree.  This  equation  will  have  four 
roots ;  there  will  therefore  be  four  sets  of  coordinates,  all  four 
of  which  may  be  real,  two  real  and  two  imaginary  (since  imag- 


GENERAL  EQUATIONS   OF   CONIC   SECTIONS. 


129 


inary  roots  enter  in  pairs),  or  all  four  imaginary,  and  in  any 
case,  since  equal  roots  occur  in  pairs,  these  four  sets  may  reduce 
to  two. 

When  two  sets  of  values  reduce  to  one,  that  is,  are  equal,  two 
of  the  points  of  intersection  become  coincident  and  the  conies 
are  said  to  touch  each  other  at  that  'point.  Hence  two  conies  can 
touch  each  other  at  but  two  points. 

The  several  cases  are  illustrated  in  the  figure.  1  and  2  inter- 
sect in  four  points,  all  four  sets  of  values  of  x  and  y  being  real ; 


Fig.  68. 


1  and  3  intersect  each  other  in  two  points  and  touch  each  other 
in  one,  two  sets  of  values  being  equal ;  1  and  4  touch  each  other 
in  two  points  ;  1  and  5  have  no  points  in  common,  the  roots 
being  all  imaginary  ;  while  1  and  6  intersect  in  two  points, 
two  sets  of  values  being  real  and  two  imaginary. 

If  the  conies  are  circles,  the  simplest  way  of  combining  their 
equations  is  by  subtraction.     Thus,  let  (Art.  48) 

f^x'  +  D!/  +  Ex-hF=0,  (1) 

jr-  +  a--  +  D'u  +  E'x  +  F'  =  0,  (2) 

be  the  given  circles.     Subtracting, 


130  ANALYTIC   GEOMETRY. 

{D-D')y  +  {E-E')x+{F-r)  =  ^,  (3) 

from  which  we  may  find  the  value  of  either  variable  in  terms  of 
the  otlier,  and  substituting  it  in  either  (1)  or  (2)  find  that  ol 
the  other. 

Cor.  1.  Since  (3)  is  of  the  first  degree,  two  circles  can  inter- 
sect each  other  in  but  two  points,  and  hence  can  touch  each 
other  but  in  one. 

Cor.  2.  Representing  the  first  members  of  (1)  and  (2)  by 
S  and  S\  then  S  +  kS'  =  0  is  a  locus  passing  through  all  the 
points  of  intersection  of  (1)  and  (2)  (Art.  37) .  When  A-  =  —  1, 
that  is,  when  the  equations  are  subtracted,  the  resulting  equa- 
tion, (3),  is  of  the  first  degree  ;  hence  (3)  is  the  equation  of  the 
chord  common  to  the  two  circles. 

Examples.     Find  the  intersections  of  the  following  loci : 

1.  /  +  .^■2  =  25,  y'  =  ^ix.  Ans.   (3,  ±4). 

2.  y-=\Qx-x\  y'  =  -2x.  Ans.    (0,0),   (8,  ±4). 

3.  y-  +  Ax''=2b,  4/ -  25.^-'  =  - 64. 

Ans.   (2,  ±3),  (-2,  ±3). 

4.  2/2  +  :^  -  3?/  +  2.x-  -  7  =  0,  y-  +  x"  -  3y  +2x  -f-  1  =  0. 

Ans.    Concentric. 

5.  y2^^i_^y_2x  +  l  =  0,  y-  +  x--\-3y-4.x  +  3  =  0. 

Ans.   (1,  0)  ;   (|,  -i). 

6.  y^-Sy  +  8x  +  lO  =  0,  y--\-4x  +  6  =  0. 

Ans.    (-1,  -2);   (-|,  -1). 

7.  Sy^  +  2a^-6y  +  8x-10  =  0,    3y- +  23^+ 6x-4:  =  0. 

8.  3?/-+ 2x2- 6?/+ 8a; -10  =  0,    3^/2+ 2ar^- 6?/+ 8a;  +  1  =  0. 

9.  Prove  that  if  two  circles  intersect,  the  common  chord  is 
perpendicular  to  the  line  joining  their  centres. 

Let  »/2+r2=/J2,  and  (</ -  n)2  +  (r- m)2=  i?,2  be  the  circles.      Sub- 
tracting these  equations,  the  equation  of  the  common  chord  is 
2nu  +  2mx^n^--m^=R^-Ri^,     or     ,j^-"'.r+C, 


f} 


GENERAL   E(,)UATIONS    OF    CONIC    SECTIONS. 


131 


in  which  C  represents  the  absolute  term.     The  line  througli  the  centres  is 


y  =  —  x. 
Ill 


10.  Prove  that  the  perpendicular  from  the  centre  of  a  circle 
on  :i  chord  bisects  the  chord. 


100.  Defs.  Conies  having  the  same  eccentricity  are  said  to 
be  similar.  If  the  corresponding  axes  are  also  parallel,  each 
to  each,  they  are  said  to  be  similar,  and  similarly  placed. 

101.  All  conies  in  whose  equations  the  terms  of  the  second 
degree  are  the  same  are  both  similar  and  similarly  2)lciced. 

Let  A>f  +  Bxij  +  Cx-2  +  B  'y  +  E'x  +  F'  =  Q,  ( 1 ) 

A}/  +  Bxy  +  Ox'  +  />"//  +  E"x  +  F"  =  0,  (2) 

be  the  equations  of  the  conies,  the  coefficients  of  the  first  three 

terras   l)eino;   the  same.     We   have    seen   that   tan2v  = 

.  ^      A-C 

(Art.  81),  in  which  y=  the  angle  made  by  the  axis  of  the  conic 

with  X.     Since  y  depends  only  upon  A,  jB,  and  C,  the  conies 
are  similaily  placed. 

The  axes  of  the  conies  being  then  parallel  to  each  other,  we  \ 
may  transform  to  a  system  of  reference  whose  axes  are  parallel 
to  those  of  both  curves  ;  and  this  transformation  does  not 
change  the  coefficients  A  and  C  (Art.  81,  Cor.  1),  and  is  pos- 
sible only  when  B  is  the  same  in  both  equations.  Transforming 
now  each  equation  separately  to  an  origin  at  the  centre  of  the 
corresponding  conic  and  parallel  axes,  since  this  transformation 
does  not  change  the  values  of  A  and  C  (Art.  84,  Cor.  3),  the 
equations  become 

Ay-  +  Cx-  +  i^i  =  0,  Ay'  +  Cx-  +  i^o  =  0, 

and  the  squares  of  the  semi-axes  are  respectively 

-t^l        1,(2 -^1        --^1        .(19  J^  •>        -Lll-y  Fr, 


a'-  = 


t,f2 


h'-  =  --K  and  a'"  = 


A 


& 


b"'  = 


A 


,^2 


Now  e^  =  1  ±  "--  =  1  ± 


fj 


—-,  or  the  eccentricity  is  the  same  for 

«'-  a"'' 

each  conic ;  hence  they  are  also  similar.         /  /-e^j'   ^ 


*  fi 


f-^^' 


/ 


■^■■^•^  t-w 


i- 


''^  .^.^.J,      3.^1^  £0^^ 


/}, 


/  c^  c. 


132  ANALYTIC    GEOMETRY. 

Cor.  1.  All  parabolas  are  similar,  since  e  =  l  for  every 
parabola. 

Cor.  2.     All  circles  are  similar  aud  similarly  placed. 

Cor.  3.  If  two  couics  are  similar  and  similarly  placed,  they 
can  intersect  each  other  in  but  two  points  and  touch  each  other 
in  but  one.  For,  subtracting  the  equations  (1)  and  (2),  we 
have  {D'-D")y^{E' -E")x+F' -F"  =  Q,  which  is  a 
straight  line  passing  through  all  the  points  of  intersection  of 
(1)  and  (2).  Combining  this  equation  with  (1)  or  (2),  there 
results  two  sets  of  coordinates,  which  may  become  equal.  The 
above  equation  is  the  equation  of  the  common  chord,  or  tangent. 

Cou.  4.  If  two  conies  differ  only  in  their  absolute  terms, 
they  are  concentric. 

102.  To  find  the  condition  that  an  equation  of  the  second 
degree  may  represent  two  straight  lines. 

We  have  seen  that  two  intersecting  straight  lines  is  a  par- 
ticular case  of  the  hyperbola  (Art.  72),  and  that  two  parallel 
straight  lines  is  a  particular  case  of  the  parabola  (Art.  85).  It 
is  further  evident  that  if  we  multiply,  member  by  member,  two 
equations  of  the  form  ay  +  hx  -)-  c  =  0,  there  will  result  an 
equation  of  the  second  degree,  and  that  this  latter  will  represent 
the  two  straight  lines  represented  by  the  factors,  for  it  will  be 
satisfied  by  the  values  of  the  coordinates  which  make  either  of 
its  factors  zero.  Conversely',  if  an  equation  of  the  second  degree 
can  he  resolved  into  two  factors  of  the  first  degree,  it  will  rejvesent 
both  the  straight  lines  represented  by  these  factors.  Sometimes 
these  factors  can  be  discovered  by  simple  inspection.  Thus, 
xy  =  0  can  be  resolved  into  the  factors  a;  =  0,  y  =  0,  and,  since 
these  are  the  equations  of  the  axes,  xy  =  0  is  the  equation  of 
both  axes.  Again,  x- —  y- =  0  can  be  resolved  iuto.r  +  v  =  0, 
x  —  y  =  0,  which  are  the  bisectors  of  the  angles  between  the 
axes,  and  therefore  a*^  —  y-  =  0  is  the  equation  of  both  bisectors. 
As  these  factors  are  not  always  readily  seen  on  inspection  of  the 
equation,  it  becomes  desirable  to  determine  the  general  condi- 
tion for  the  existence  of  two  factors  of  the  first  degree. 


GENERAL   EQUATIONS    OF    CONIC    SECTIONS.  133 

Let  Af  +  B.vy  +  Cx"  +  Dij  +  Ex  +  i^ = 0 

be  the  general  equation  of  a  conic.     Solving  it  for  3/, 


In  order  that  tliis  equation  may  be  capable  of  reduction  to 
the  form  // =  ax  ±  ft,  the  quantit}'  under  the  radical  must  be  a 
perfect  square.  But  the  condition  that  the  radical  should  be 
a  perfect  square  is 

{BT-  -  4rAC)  (I>-  -  4  AF)  =  {BD-2  AE)\ 

Expanding  and  reducing, 

4.ACF  +  BDE  -  AE-  -  CD'-  -FB'  =  0, 

which  is  the  required  condition.  If  the  coetiicieuts  of  the  given 
equation  satisfy  this  relation,  the  equation  represents  two 
straight  lines  ;  to  find  the  lines,  solve  the  equation  for  y  and 
extract  the  root  indicated  by  the  radical. 

Examples.      Determine    which   of    the    following    equations 
represent  pairs  of  straight  lines,  and  find  the  lines  : 

1.  Ay'-oxi/-hx'  +  2y  +  x-2=0. 

Applying  the  test,  we  find  -  32  -  10  -  4  -  4  +  50  =  0. 
To  find  the  lines,  solving  for  y,  we  obtain 

1/  =  ^il^  ^  Vox^-  3(j.r  +  36  =  ^"^  ~  ^  ±  -  (.Sx  - 6). 

•^8  8  8  8^^ 

Hence  the  lines  are  ^  =  .r  —  1  and  y  —  I  x  +  h 

2.  'dy--8xy  +  '3x-  1=0. 

.3.  y-  -  2y  -  .r  +  1  =  0.     Ans.  y  —  x  -  1  =0,  y -j- x  -  I  =0. 

4.  xy  —  ay  —  bx -\-(ib  =  0.  Ans.  x=a,  y=h. 

5.  y-  +  -ixy  +  -ix-  —  \  =  Q.  Ans.  y  -\-2x=±2. 

6.  xy  +  x-  —  y  +  ^x-lO  =  0. 

Ans.  ic  —  1  =  0,  ^  +  a;  +  10  =  0. 


134 


AN^UiYTIC   GEOMETKY. 


SECTION  IX.— TANGENTS  AND  NORMALS. 


103.  Defs.  Let  M3I'  be  any  locus  and  AB  any  secant  cut- 
ting the  locus  in  the  points  P'  and  P".  If  AB  be  turned  about 
P'  regarded  as  fixed,  till  P".  moving  in   the  locus,  coincides 

with  P',  AB  will  then  have  but 
one  point  in  common  with  the 
locus  and  is  called  the  tangent 
at  that  point. 

The  direction  of  the  tangent 
is  that  in  which  the  generating 
X      point  is  moving  as  it  passes 
through  the  point  of  tangency, 
or  the  slope  of  the  locus  at  any 
point  is  the  slope  of  its  tan- 
Fig.  69.  gent  at  that  point.     The  per- 
pendicular to  the   tangent    at 
the   point  of  tangency,  lying  iu  the  plane  of  the  curve,  P'A^, 
is  the  normal. 

104.  General  equations  of  the  secant  and  tangent  to  a  conic. 


Y 

^P^ 

/^ 

A^ 

0/ 

/     \ 

\ 

/ 

\ 

\^ 

X^/' 

Let 


.'/  -  ?J  = 


.'/  —y 


•^       x'  —  X 


J,{X-X') 


(1) 


be  the  equation  of  a  straight  line  passing  through  any  two  given 

points  (ic',  ?/'),  (x",y").     The  coefficient  of  x  when  the  cqua- 

?y'  -  y"  y'  -  y" 

tion  is  solved  for  ?/  being  '—. — '—r.^  we  have  "— ; — '—r,  =  a  =  the 

*  x'  —  x"  X  —  x" 

slope  of  the  line.     Also,  let 

/(.x,?/)  =  0  (2) 

be  the  equation  of  any  conic.     If  the  two  points  through  which 
the  given  line  passes  are  on  the  conic,  we  must  have 


TANGENTS    AND   NORMALS.  135 

/(x',2/')=0,   aud  f{x",y")=0. 

y'  -y" 

Hence,  if  we  form  —, ^,  from  these  two  equations  aud  sub- 

stitute  the    vahie   thus   found  in   ( 1 ) ,    we  shall    introduce  the 

condition  that  (1)  is  a  secant  of  (2). 

?/'  —  ?/" 
Representing  this  value  of '  ,      '_,,  by  a,,  the  equation  of  the 

secant  will  be 

.'/-?/'  =  »,  (;f  — a-'). 

If,  now,  in  the  value  of  a,,  we  make  x"  =  x'  and  y"  =  y',  that 
is,  suppose  the  point  {x",y")  to  coincide  with  (x',y'),  the  secant 
will  become  a  tangent ;  hence,  representing  what  a^  becomes 
under  this  supposition  by  (',,  the  equation  of  the  tangent  will  be 

y  -y'  =  a,  (x  -  a;') , 

in  which  (x',  y')  is  the  point  of  tangency. 

Examples.    1.    Equation  of  the  tangent  to  the  circle 

y-  +  X-  =  R-. 

Let  (.(',  ij'),  (j",  I/")  be  any  two  points  of  the  circle.     Then 

y'-  +  x>^  =  R^,     and     ij"'^  +  x"^  =  R^. 

Subtracting,  we  liave 

y'--^"-+-c'--x"-!=0,  or  (//'-//")  (.</'  +  y")  =  - (x' -  x")  (x'+x"), 

,                                            ?/'  —  //"          x'  +  x" 
whence ■■^—  = 

x'  —  x"        y' +  y" 
Substituting  this  value  in  the  equation  of  a  line  through  two  given  points, 


it  becomes  y  —  y'  —  —  ' — I!— — (^x  —  x'), 

.'/'  +  !/" 

the  general  equation  of  the  secant  line  to  the  circle.     Making  now  x"  =  x' 

x' 
and  y"  =  //',  we  have       y  —  y'  = (-'"  —  x'), 

y' 

for  the  equation  of  the  tangent.     This  equation  may  be  simplified  by  clear- 
ing of  fractions  and  replacing  //'-+  .r'2  by  its  equal  /?-;  whence,  finally, 

yy'^rx'^.R\ 


136  ANALYTIC    GEOMETRY. 

the  general  equation  of  the  tangent  to  the  circle  tj- +  x-  =  R^,  (x',  //')  being 
the  point  of  tangency. 

Note.  The  process  is  the  same  whatever  the  equation  of  tlie  conic; 
that  is,  whatever  its  species  or  the  axes  of  reference ;  and  the  student 
should  thoroughly  master  the  above  illustration  as  exemplifying  a  method 
for  producing  the  equation  of  a  tangent  to  any  conic  when  referred  to  a 
rectilinear  system.  Thus,  if  the  circle  be  referred  to  a  diameter  and  the 
tangent  at  its  left-hand  vertex,  its  equation  is 

Hence  ij'2=2Rx' -  x'-     and     //"^=  2i?x"-x"2; 

and,  by  subtraction,     y'^  -y"-^  =  2R(x'-  x")  -  (x'^  -  x"^)  ; 

whence  y^^2R -jx' ^x'') , 

x'  —  x"  y' +  y" 

which  becomes  when  the  points  coincide.      The  equation  of  the 

y  R  —  x' 

tangent  is  therefore  y  —  y'  = ; —  (x  —  x'). 

y' 

2.  Find  the  equation  of  the  tangent  to  the  elUpse 

a^y^ -{-b^x^  =  a^b^.         Ans.  a-yy' -j-b'-xx' =  a-b-. 

If  a=zb,  this  becomes  yy'  +  xx'  =  a-,  or  R'-,  the  tangent  to  the  circle,  as 
above.  Since  the  equation  of  the  liyperbola  differs  from  that  of  the 
ellipse  only  in  the  sign  of  b-,  we  have  also  tlie  equation  of  the  tangent  to 
the  hyperbola,  n'^  yy'  —  b^xx'  =  —  a'-b'-. 

3.  Deduce  the  equation  of  the  tangent  to  the  hyperbola  by 
the  general  process. 

4.  Find  the  equation  of  the  tangent  to  the  parabola  y-  =  2px. 

Ans.  yy'  =  p  (x  +  x') . 

When  the  central  equations  of  the  ellipse  and  hyperbola  in 
terms  of  the  semi-axes  are  used,  and  the  equation  of  the  parabola 
referred  to  its  axis  and  vertex,  the  corresponding  equations 
of  the  tangents  are  easily  romcmbcred  from  the  fact  that,  by 
dropping  the  accents  which  distinguish  the  coordinates  of  the 
point  of  tangency,  they  become  the  equations  of  the  curves 
themselves. 


TANGENTS  AND  NORMALS.  137 

105.  Problems.  Under  the  head  of  taooeucv  the  foUowins? 
simple  problems  occur. 

FiKST.  To  ivrite  the  equation  of  a  tangent  at  a  given  point  of 
a  conic,  and  to  find  the  slope  of  the  conic  at  that  p)oint. 

Find  the  general  equation  of  the  tangent  to  the  conic  by  the 
preceding  method,  and  substitute  in  tliis  equation  for  .«',  y\  the 
coordinates  of  the  given  point.  The  coefficient  of  x  in  the  result- 
ing equation,  when  it  is  put  under  the  slope  form,  will  be  the 
required  slope.  Thus,  the  tangent  to  the  circle  if  +  x}=  100  at 
the  point  (—  (»,  —  8)  being  required,  make  x'  =  —  G,  y'  =  —  8 
in  the  equation  of  the  tangent  yy'  ■}-  xx'  =  R-,  R  being  10,  and 
we  have  —  8v/ —  6.r  =  100,  or  4?/ +  3.T  + 50  =  0,  which  is  the 
tangent.  Solving  for  y,y=  —  f.'c  — ^,  or  aj=  —  |,  and  the 
angle  which  the  tangent  makes  with  X=  tan~^f . 

Second.  To  find  the  point  on  a  conic  at  ivhich  the  conic  has  a 
given  slope. 

In  this  case  the  coordinates  of  the  point  of  tangency  are 
unknown.  To  find  them  we  have  the  two  equations  f{x',  y')  =0 
(since  tlie  point  is  on  the  conic),  and  the  given  condition 
a^  =  a',  where  a'  is  the  given  slope.  Combining  these  equations 
we  find  x',  y',  the  required  poiut  of  tangency.  If  more  than  one 
set  of  values  for  .^•',  y\  are  found,  there  is  more  than  one  solu- 
tion. If  the  value  of  either  x'  or  y'  proves  to  be  imaginary, 
there  is  no  poiut  fulfilling  the  condition.  Thus,  at  what  point 
of  the  ellipse  dy^  +  4.r  =  36  does  the  tangent  make  an  angle  of 

Ir  r'  4  x' 

45°  with  X?     By  condition,  a^  = -^  = ,  =  1  •     Also 

a-y  9  y 

9y"  +  ix'-  =  S6. 

Substituting  x'  =  —  ^y'  fvom  the  first  in  the  second,  we  obtain 
9  yi2  +  4»2/'^  =  36,  or  y' =  ±  ¥^'  Hence  x'  =  ^:  ^^3;^  -There 
are  therefore  two  points  at  which  the  slope  is  1  ;  one  in  the 
second  angle,  (  — |s-?X/^,  V2),  the  other  in  the  fourth, 

7^   ^    ''(1V2,    -  V-2). 


138  ANALYTIC   GEOMETRY. 

Third.    To  find  the  equation  of  a  tangent  to  a  conic  ivhich 
passes  through  a  given  point  without  the  curve. 
Let  /(.1-,  ?/)  =  0  be  the  ('(luation  of  the  conic,  and 

that  of  the  tangent.  In  this  case  also  the  coordinates  of  the 
point  of  tangency  are  unknown.     To  find  thorn  we  have 

/(a;',  ?/')  =  0  (since  the  point  of  tangency  is  on  the  conic), 

and  c^  (7i.  A-,  x\  ?/')  =  0,  in  which  //,  A-,  are  the  coordinates  of  the 
given  point  (since  the  tangent  passes  through  it).  Combining 
these  equations,  we  find  x'  and  ?/',  and  there  will  be  as  many 
solutions  as  there  are  found  sets  of  values  for  x\  y'.  If  either 
x'  or  ?/'  should  prove  imaginary,  the  problem  is  impossible. 
Thus,  to  find  the  equation  of  the  tangent  to  the  circle  i'/^+  x'=  25, 
passing  through  (7,  1).  Since  the  point  of  tangency  is  on  the 
circle,  y'-  +  x'-=2b.  Since  the  point  (7,  1)  is  on  the  tangent 
yy'  +  xx'  =  R-,  we  have  also  ?/'  +  7a''  =  25.  Combining,  Ave 
find  x'  —  3,y'=  4,  and  x'  =  4,  ?/'  =  —  o.  There  are  therefore 
two  tangents  to  the  circle  through  (7,  1),  namel}',  ■iy-{-3x  =  25, 
and  Ax  —  3y  =  25. 

Cor.  Since  the  equation  of  a  conic  is  of  the  second  degree, 
and  that  of  the  tangent  of  the  first  degree,  no  more  than  two 
tangents  can  be  drawn  having  a  given  slope,  or  through  a  given 
point  without  the  conic. 

Examples.  1 .  Find  the  equation  of  the  tangent  to  the  circle 
2/2  -\-x^=  25  at  ( -  .'},'-l ) .  A71S.  4//  -  3a;  =  25. 

2.  Find  the  slope  of  the  circle  y- -\- x- =  R-  at  the  points 
whose  abscissas  and  ordinates  are  numerically  equal. 

Ans.  45°;   135°. 

3.  Find  the  equations  of  the  tangents  to  the  circle  y'^+  x-=  1 OU 
passing  through  the  point  (10,  5).   Ans.  4?/  +  3.i-  =  5U;  .t=10. 

4.  P'ind  the  slope  of  the  ellipse  3?/-  +  .t'-  =  3  at  the  points 
a;'  =  0;  y'  =  Q\  x' =  f  •  Ans.  0°;  1)0°;   135°  a«d  225°. 


TANGENTS   AND   NORMALS.  189 

5.    Find  the  points  on  the  ellipse  8?/-  +  4ar  =  32  at  which  tlie 

ith  X. 

4         2  \     /       4  2 


tangent  makes  an  angle  of  135°  with  X. 


A 


US. 


~5 


.V3     V3y     V     V3         V3 

6.  Find  the  tangents  to  the  ellipse  20y^  -\-9af  =  324  passing 
through  (— 1,  6).  Am.   5?/  + 3a;  =  27;  35^  —  33x'=  243. 

7.  Write  the  equations  of  tangents  to  the  parabola  y-  =  8x 
at  the  points  x'  =  8  ;  x'  =  2. 

8.  Find   the  point  on    9?/- —  4a;- =  —  3G    where  the  tangent 
makes  an  angle  with  X  whose  tangent  is  ^-     Aris.  No  siicJi  point. 

9.  Sliow  that  the   focal  tangent  to  the  parabola  niakes  an 
angle  of  45°  with  X.      (Find  the  slope  at  y'  —p.) 

10.  Find  the  eccentricity  of  the  ellipse  25?/- +  9  a;- =  225,  by 
finding  the  slope  at  the  extremity-  of  the  parameter.         Ans.  4- 

11.  Show  in  the  same  manner  that  the  eccentricity  of  the 
hyperbola  16?/-  —  9a;-  =  —  144  is  f- 

12.  Toicrite  the  equation  of  the  tangent  to  the  ellipse  in  terms 
of  the  slojye.     The  equation  of  the  tangent 

is  a-i/ij'  +  b-x.r'  =  a%'^,  or  //  = —  r  -\ Let ^  =  m  =  slope  of  the 

a'-J/'         .'/'           1,2      «-.'/' 
tangent,  whose  equation  then  becomes  y  =  ?nr  H To  eliminate  y',  we  have 

Ifix'  =  -  a^i/'m,     and     a~y'-^  +  b-x'-  =  a~b- ; 
whence  a^t/'-  +  1^Il!z!^  =  a^h^^     or     y'^  (ahn^  +  b'^)  =  b\ 

b-^ 


from  which  we  obtain  --  =:  Va-m'^  +  b'^.     Thus  the  equation  of  the  tangent  is 

.!/  

y  =  771 X  +  •s/ahn'^  +  b"^- 

Changing  the  sign  of  b-,  and  making  a  =  b,  we  have  the  corresponding 
equations  for  the  hyperbola  and  circle, 


y  =  mx  +  \/a-m^  —  b'^,     and     y  =  inx  +  a  Vm'^  +  L 

13.    To  ivrife  the  equation  of  the  tangent  to  the  parabola  in 

terms  of  the  slope.  p 

Ans.  y  =  mx  4-  -^- 
^  2  m 


140  ANALYTIC    GEOMETRY. 

14.  The  rectangle  of  the  perpendiculars  from  the  foci  of  an 
elUiJse  upon  the  tangent  is  constant  and  equal  to  the  square  of  the 
sem i-conjugate  axis. 

Putting  the  equation  of  tlie  tangent  under  the  normal  form,  we  have 

n^i/y'  +  b^xx'  —  a-b-  _  ^ 

Substituting  in  succession  tlie  coordinates  of  the  foci,  ae,  0,  and  —ae,  0, 
for  X  and  y,  and  taking  the  product  of  the  results,  we  have,  jmtting  the 
radical  =  D  for  brevity, 

-  (b^aex'  -  a%^)  {b^aex'  +  a^/)"^)  _  a'^b^  -  b^a^e-x'^ 

^  bHaW^+h^^'-)  ^  1-2 
Z)2 
This  property  is  true  also  of  the  hyperbola. 

15.  The  'perpendicular  from  the  focus  of  an  hyperbola  upon 
the  asymptote  is  equal  to  the  semi-conjugate  axis. 

This  may  be  regarded  as  a  particular  case  of  the  foregoing,  the  asymp- 
tote being  a  tangent  whose  point  of  contact  is  at  an  infinite  distance.  Or, 
directly,  the  equation  of  the  asymptote  y -■  -  x  under  the  normal  form  is 
a  //  —hx  _ 


0 ;  substituting  the  coordinates  of  the  focus 


x  =  ae  =  va-  +  b^,     y  =  U, 
this  expression  reduces  to  —  b. 

16.  To  find  the  length  of  a  tangent  from  a  given  point  loith- 
out  a  circle. 

Let  (xi,  y■^)  be  the  given  point  Pj,  and  P'  the  point  of  tangency, 
(x —  my+ (y —ny^  —  R^—0  being  the  equation  of  the  circle  and  C  its 
centre.  Then,  since  the  radius  to  the  point  of  contact  is  perpendicular 
to  the  tangent,  P,P'-^=  P.C^  -  CP'-\  But  P,C''=(x^-my+(y,-ny 
(Art.  7),  CP'-  =  R-.     Hence 

PiP'2=  (a-i-  ?n)2+  (yj_  n)2-i?2. 

Now  this  is  what  the  equation  of  the  circle  becomes  when  the  coordinates 
of  the  given  point  are  substituted  for  .r  and  y ;  hence,  put  the  equation 
of  the  given  circle  under  the  form  /(.r,  ,//)  =  0,  and  substitute  for  x  and  y 
the  coordinates  of  the  given  point.  The  result  will  be  the  square  of 
the  required  distance.  Thus  the  length  of  the  tangent  to  the  circle 
y24.x2_6i/  +  8x-ll  =  0  from  (5,  1)  is  Vf+^S - 6  +  40 -  11  =  7. 


TANGENTS  AND  NORMALS.  141 

17.  If  hvo  circles  are  tangent  internally  and  the  radius  of  the 
larger  is  the  diameter  of  the  smaller,  all  chords  of  the  larger 
through  the  point  of  contact  are  bisected  by  the  smaller. 

Take  the  origin  at  the  point  of  contact  and  the  diameter  as  the  axis  of 
A'.  Then  the  equation  of  any  chord  is  y  =  «.r,  and  the  equations  of  the 
larger  and  smaller  circles  are  y^=  2  Rx  —  .r-  and  y-  =  Bv  —  x^,  respectively. 

The  chord  intersects  the  former  at  (  ,    —  ]  and  the  latter  at 

R  Ra   \  V«''  +  l     «'  +  l^ 


a2+l     a2+l/  / 


106.  Chord  of  contact.  Tangents  are  drawn  to  a  conic  from- 
a  given  external  point ;  to  Jind  the  equation  of  the  chord  of  con- 
tact. 

First.  The  ellipse  and  hyperbola.  Let  (/i,  k)  be  the  external 
point,  and  {x\y'),  {x'\y"),  the  points  of  tangency.  Then, 
since  both  tangents  pass  throngh  (h,  k).,  these  coordinates  must 
satisfy  their  equations  ;  or 

a'ky'  ±b-hx'  =±aV)\  (1) 

a-ky"  ±b-hx"  =  ±a-b-.  (2) 

Then  a^ky    ±lrhx   =  ±  fr6'-^ 

is  the  equation  of  the  chord;  for  it  is  satisfied  by  (x',  ?/'), 
(x",  y"),  as  sliown  by  (1)  and  (2),  and  is  of  the  first  degree 
with  respect  to  x  and  y,  and  therefore  represents  a  straight  line 
through  {x\  y') ,  {x",  y") . 

CoR.    The  chord  of  contact  to  the  circle  is  ky  +  hx  =  R^. 


Second.    The  parabola.     The    equations  of   both    tangents 
must  be  satisfied  for  (/i,  k)  ;  hence 

ky'  ^p{h  +  x'), 

ky"^p{h  +  x"). 

Then  ky  =  p  (Ji  +  x) 

is  the  chord  of  contact,  the  reasoning  being  identical  with  that 
above. 


^ 


142  ANALYTIC    GEOMETRY. 

It  will  be  observed  that  the  equations  of  the  chord  of  con- 
tact are  derived  from  those  of  the  tangent  by  changing  the 
coordinates  of  the  point  of  tangency,  x\y\  into  those  of  the 
external  point ;  these  equations  are  therefore  easily  memorized. 

107.  General  equation  of  the  normal  to  a  conic.  The  equa- 
tion of  any  line  through  the  point  of  tangency  {x',  y')  is 

y-y'  =  a  {X  -  X')  . 

But  the  normal  is  perpendicular  to  the  tangent,  hence  a  must 
equal ,  ««  being  the  coefficient  of  x  in  the  equation  of  the 

tangent  when  under  the  slope  form ;  or  the  equation  of  the 
normal  is 

y-y'  =  -^(x-x'). 

a. 
Examples.    1.    Find  the  equation  of  the  normal  to  the  circle 
y-  +  X-  =  R-. 

r'  R~ 

The  tangent  to  the  circle  is  iiij'  +  xx'  =  R^,  or  i/-  —  '-.r-\ — -;  hence 

/  '         '  !^.  '> 

at=  —  —,  and  the  normal  is  ^  —  ;/'=  ■—  (r—  .f ') ;  or,  clearing  of  fractions, 

x'y  —  y'x  =  0.  Since  this  equation  has  no  absolute  term,  the  normal  passes 
through  the  origin,  which  is  the  centre ;  hence  the  normal  to  a  circle  is  the 
radiux  to  the  point  of  tangency. 


2.  Find  the  normal  to  the  ellipse  ahf-  +  b'x^  =  a^b^. 

b^x' 

3.  Find  the  normal  to  the  hyperbola  a-y^  —  b^a?  =  —  a?b-. 


Ans.  y  -  y'  =  -^,(x-  x') . 


Ans.  y-y'  =  -%^,(^-^-" 


b^x 

•1  •    Find  the  normal  to  the  parabola  y^  =  2px. 

v' 
Ans.  y-  y'  =  —  ^-  {x  —  x'). 


5.    Find  the  normal  to  the  circle  y^  =  2  Ex  —  .r 


A71S.  y-y'  =  ^^^{x-x'). 


TANGENTS   AND   NOUMALS. 


143 


6.    Write  the  equation  of  :i  uormal  in  tlie  following  cases  : 

(a)  to  a  circle  whose  radius  is  o  at  the  point  (o,  —  4). 

(b)  to  an  ellipse  whose  axes  are  6  and  4  at  the  point  x'  =  1. 

(c)  to  a  parabola  whose  parameter  is  9  at  the  point  x'  =  4. 

(d)  to  an  hyperbola  whose  axes  are  6  and  4  at  the  point  .t''=8. 
'3?/  +  4.b-=0;  3?/  =  ±  9  V2a;  ip  5  V2  ; 

3  //  =  q:  4 .«  ±  34  ;  48^/  =  q:  9  Vo5  iK  ±  104  V55. 


Ans. 


108.  Defs.  That  portion  of  the  axis  of  X  intercepted 
between  the  ordinate  from  the  point  of  tangency  and  the  tan- 
gent is  called  the  subtangent.  In  like  manner  that  portion  of 
the  axis  of  X  intercepted  between  the  ordinate  and  the  normal 
is  called  the  subnormal.  Thus  (Fig.  70) ,  T3f  and  MN  are  the 
subtangent  and  subnormal  to  the  point  P'.  4'' 

109.  To  Jind  the  subtangent  and,  subnormal  at  any  point  of 
a  conic. 

Let  Xi=  OT  represent  the  intercept  of  the  tangent  on  the  axis 
of  X ;  that  is,  the  value  of  x  when  y  is  made  zero  in  the  equation 
of  the  tangent.     Then,  x'  being  the 
abscissa  OM  of  P',  the  point  of  tan- 
gency, 

TM=  subtangent  =  OM  -0T=  x'  -  x,. 

Similarly, 

lfiV=  subnormal  =  ON—  OM=x^—x', 

Xn  being  the  X-intercept  of  the  nor- 
mal, or  the  value  of  x  when  y  is  made 
zero  in  the  equation  of  the  normal. 

Examples.   1.  To  find  the  subtangent  of  the  ellipse  and  the 

hyperbola. 

The  equations  of  the  tangents  are  ar yij'  ±  h-.rx'  =  ±  d-b'^.     When  y  —  0, 

X  —  Tt=  —  for  both  curves.     Hence,  also,  for  both  curves, 
.r' 

'■■-a.2  ^ 


subt.  =  x'  —  .Tj  —  : 


.' 


144 


ANALYTIC    GEOMETRY. 


Cor.   Since  a;,  =  — , ,  x, :  a  :  :  a  :  x' ,  or  OT :  OA' :  :  OA' :  03L 
x' 

Hence,  the  semi-transverse  axis  is  a  mean  proportional  between 
the  intercejits  of  the  tangent  and  the  ordinate  of  the  point  of  tan- 
gency.     (Figs.  71  and  72.) 

This  principle  alTorcls  a  method  of  constructing  a  tangent  at 
any  point. 

First.  The  ellipse.  Let  P  be  tlie  point.  Describe  the  circle 
AP'"A'    on    the    transverse    axis,    and  produce    the   ordinate 


Fig.  71. 


through  P'  to  meet  the  circle  at  P'".  At  P'"  draw  the  tangent 
to  the  circle,  P"T.  Then  PT  is  the  required  tangent.  For, 
from  the  right  similar  triangles,  OP'" 31,  OP"'T, 

OT:  OP"'{  =  OA')  :  :  OP'":  031; 


or 


Since  both  OT=x,=  %  and  31T  =  suht.  =  ^—^^ ,  areinde- 

x'  x' 

pendent  of  b,  tangents  to  all  ellipses  having  the  same  transverse 

axis,  at  points  having  the  same  abscissa  x'  =  031,  will  evidentlv 

pass  through  T. 


Second.  The  hyperbola.  Let  P'  be  the  point.  Draw  the 
ordinate  P'J/,  and  on  AA',  03r,  as  diameters,  describe  circles 
intersecting  at  Q.     Draw  QT  perpendicular  to  X.     Then   TP' 


TANGENTS    AND   NORMALS. 


145 


is  the  required  tangent.     For,  joiuiug  Q  with  0  and  M,  from 
the  similar  triangles  0Q3f,  OQT,  we  have 

OM:OQ(  =  OA')  ::0Q:  OT, 


or 


x^:  a  :  :  a  :  X 


Fig.  72. 


/^ 


Cor.  Since  0T=  ^,  0 Twill  be  zero  only  when  x'  =  cc  ;  that 

x' 

is,  when  the  tangent  coincides  with  the  asymptote  (Art.  91). 

2.  To  find  the  subtangent  of  the  parabola. 

Ans.  Xf  =  —  x';  subt.  =  2a;'. 

It  appears  from  this  result  B^ 

that  the  subtangent  of  the  pa- 
rabola  is  bisected  at  the  vertex. 
Hence  to  construct  a  tangent 
at  a  given  point  P',  draw  the 
ordinate  P'J/and  make 

0T=  OM. 

Then  TP'  is  the  required  tan- 
gent. 

Fig.  73. 


146  ANALYTIC    GEOMETKY. 

3.  To  find  the  subnormal  of  the  parabola. 

The  equation  of  the  normal  being  y  —  ^'  =  —  —  (x  —  r'),  we  have,  when 

^  =  0,  X  —  Xn  =  p  +  x'.  Hence  subn.  =  t,,  —  x'  =  p,  or  the  subnormal  of  the 
parabola  is  constant  and  equal  to  one-half  the  parameter. 

Therefore,  to  construct  a  tangent  at  a  given  pointy  as  P',  draw 
the  ordinate  P'M  and  make  MN=p.  Then  P'M  is  the  normal, 
and  P'2\  perpendicular  to  it,  is  the  tangent. 

4.  The  tangent  to  the  ])arabola  bisects  the  angle  between  the 
focal  radius  and  the  produced  diameter  through  the  point  of 
contact. 

Let  P'  be  the  point  of  contact,  F  the  focus,  and  P'D  the  diameter. 

Then  FP'  =  x'  +  |  (Art.  74,  Cor.  2).     Also  TF=TO+  0F=  x'  +  |  (Ex.  2). 

Hence  Fr=FP',  the  triangle  TFP' is  isosce\es,iind  DP' T=P'TF=FP'T. 
Therefore,  to  draw  a  tangent  at  any  point,  as  P',  draw  the  focal  ifadius  P'F, 
the  diameter  P'D,  and  bisect  their  included  angle. 

CoR.  1.       FN^  OM-  OF+MN=  x'-^  +p  (Ex.  3) ; 

.-.  FN=  x'  +f  =  FT=^FP', 

or  the  circle  described  from  the  focus  with  a  radius  equal  to  the 
focal  radius  of  any  point  j^asses  through  the  intersections  of  the 
normal  and  tangent  to  that  iioint  with  the  axis.  The  triangle 
FP'N  is  thus  isosceles,  and  FNP'  =  FP'N. 

CoR.  2.  P'FN=  FP'T+  FTP'  =  2  FTP'.  Hence,  to  draw 
a  tangent  parallel  to  a  given  line,  as  AB,  from  F  draw  FP'  mak- 
ing an  angle  with  the  axis  equal  twice  that  made  by  the  given 
line.  P'  will  be  the  required  point  of  contact  and  P'T,  parallel 
to  AB,  the  required  tangent. 

Cor.  3.  To  draio  a  tangent  through  a  given  point  without 
the  curve.  Let  A"  be  the  given  point.  Join  /r  with  the  focus, 
and  with  A"  as  a  centre  and  KF  as  a  radius  describe  a  circle 
cutting  the  directrix  in  D  and  D'.  Draw  the  diameters  throuiih 
D  and  D' ;  their  intersections  with  the  curve,  P'  and  P",  are 


TANGENTS   AND   NORMALS. 


147 


the  points  of  tangencj'.  To  prove  that  KP'  is  a  tangent,  we 
have  P'D  =  P'F  by  definition  of  the  parabola  ;  also  KF  =  KD 
by  construction.  Hence  KP'  bisects  the  angle  FP'D.  Simi- 
larly, P"K  may  also  be  shown  to  be  a  tangent  at  P".  V. 

5.  The  tangent  and  normal  at  any  point  of  the  ellipse  bisect  the 
angles  formed  by  the  focal  radii  drawn  to  the  point  of  contact. 

Since  the  tangent  P'K  is  perpendicular  to  the  normal  P'N,  we  have 
only  to  prove  that  P'X  bisects  FP'F',  or  that  FN :  FP' : :  F'N :  F'Pf. 
Now,  FP'  and  F'P'  are  the  focal  radii  a  +  ex',  a  —  ex'  (Art.  57),  respectively. 


Fig,  74. 


FN=FO+ON,  in  which  FO=ae,  and  ON  is  the  ^intercept  of  the 
normal.    Making  y  =  0  m  the  equation  of  the  normal 


we  have 

Hence 

Also 

Therefore 


!/-y 


/_ 


a^  I/' 


b'^x' 


7/  (^  -  ^'). 


x=ON  = 


a2  _  J2 


x'  =  e^x'. 


a^ 


FN  =  ae  +  e^x'  =  e  (a  +  ex') . 
F'N=F'0-ON=e(a-ex') 
FN  _  e  (a  +  ex')      F'N  _  e  (a  -  ex') 


FP' 


a  +  ex'        F'P'         a  —  ex' 


or 


FN  ^  F'N 
FP'     F'P'' 


To  draiv  a  tangent  at  any  point,  as  P',  we  have,  obviously,  only  to  bisect 
the  angle  FPF'  and  draw  P'K  perpendicular  to  the  bisector. 


148 


ANALYTIC   GEOMETRY. 


6.  The  tangent  and  normal  at  any  point  of  the  hyperbola  bisect 
the  angles  formed  by  the  focal  radii  drawn  to  the  point  of  contact. 

LetP'Tbe  the  tangent  at  P'.  We  have  to  prove  that  it  bisects  the 
angle  FP'F',  or  that  FT :  FP'  -.-.F'T:  F'P'.  The  focal  radii  FP',  F'P>, 
are  ex'  — a  and  ex'  +  a  (Art.  67),  respectively.    FT=FO  —  OT,  in  which 


Fig.  75. 


FO  =  ae  and  OT  is  the  Xintercept  of  the  tangent.    Making  ^  =  0  in  the 
equation  of   the   tangent   a^yy' —  Ifixx' =  —  a?l^,    we  have    x=zOT  =  —  ; 


hence  FT  =  ae =  —  (ex'  —  a). 

X'        X' 

n  FT      F'T 

Similarly,  F'T ^- {ex' -\- a),  and,  as  before,  J^  =  jrp,- 

To  draw  a  tangent  at  any  point,  as  P',  bisect  the  angle  between  the  focal 
radii  drawn  to  the  point. 

7.  The  principles  of  Exs.  5  and  6  afford  a  method  for  con- 
structing a  tangpnf  passing  through  a  given  point  vdthout  the 
curves.     Thus  let  K  (Figs.  74,  75)  be  the  given  point.     Join  K 


TANGENTS  AND  NORMALS.  149 

with  the  nearer  focus  F\  and  with /iT  as  a  centre  and  KF'  as  a 
radius  describe  an  arc.  With  the  farther  focus  i''  as  a  centre  and 
FH  equal  to  the  transverse  axis  as  a  radius  describe  a  second 
arc  cutting  the  first  in  H  and  Q.  Join  H  and  Q  with  the  farther 
focus  ;  the  intersections  P'  and  P"  of  FQ  and  FH  with  the  curve 
are  the  points  of  tangency.  To  prove  that  KP'  is  a  tangent, 
we  have  KH==  KF'.,  being  radii  of  the  same  circle ;  also 
P'H  =  P'F',  since  each  is  equal  to  2  a^FP',  the  upper  sign 
applying  to  the  ellipse  and  the  lower  to  the  hyperbola.  Hence 
KP'  bisects  the  angle  F'P'H  in  the  ellipse  and  FP'F'  in  the 
hyperbola. 

Cor.  If  an  ellipse  and  an  hyperbola  have  the  same  foci,  at 
the  points  of  intersection  they  have  the  same  focal  radii,  and  the 
tangent  to  the  hyperbola  is  the  normal  to  the  ellipse,  and  con- 
versely. Hence  confocal  conies  intersect  each  other  at  right 
angles. 

o  8.  Tangents  at  two  points  P',  P",  of  a  parabola,  meet  the  axis 
in  T'  and  T".  Prove  that  T'T"  =  FP' -  FP",  F  being  the 
focus. 

0  9,  Two  equal  parabolas  have  a  common  axis  but  different 
vertices.  Prove  that  any  tangent  to  the  interior,  limited  by  the 
exterior,  parabola,  is  bisected  at  the  point  of  contact. 


I   f 


lO,    *C; 


150 


ANALYTIC   GEOMETPwY. 


SECTION  X.— OBLIQUE   AXES. 


CONJUGATE    DIAMETERS. 

110.  Equation  of  the  ellipse  referred  to  conjugate  diameters. 

Let  A'A"  =  2a',  B'B"  =  2b',   be   conjugate   diameters,  the 

axes  of  reference  being  taken  as  in  the  figure.     To  transform 

the  equation 

a^y^-]-b^x^  =  a-b^  (1) 

to  these  axes,  we  have  the  formulae  (Ai't.  22,  Eq.  7) 

a;  =  iCi  cos  7  +  Vi  cos  yi,  y  =  x^  sin  y  +  ?/i  sin  y^. 

Substituting  these  in    (1),  and  omitting  the  subscripts  of  x 
and  y, 

(a^  sin2yi+  b''  cos-yO  y-+2{a^  sin  y  sin  yi+  b^  cosy  cos yi)«i/ 1 
+  (a^  sin^y  +  b^  cos^  y)  ar  =  a^  b^.  j 

But,  since  the  diameters  are  conjugate,  they  must   fulfil   the 
condition  tan y  tan yi  = ^  (Art.  90), 

Cv 


OBLIQUE   AXES.  151 

or  a^siny  sinyi  =  — 6-cosy  cosyi. 

Hence  the  coefficient  of  the  second  terra  of  (2)  is  zero,  and  the 
equation  becomes 

(a^  sin-  y,  +  U'  cos^ y,)  y^  +  (a^  sin- y  +  h-  cos- y)  x"  =  a^ b\    (3) 
Making  y=0,  we  have 


x^  =  a'-  = 


ffi-Q- . 

a-sin^y  +  &-cos^y ' 

0   7  0 

Cfb- 


andwhena;  =  0,       y-  =  b'-  =  ^^r  ,      ,  ,., 

a-  sm-yi  +  0-  cos-yi 

Substituting  from  these  equations  the  vahies  of  the  coefficients 
of  y-  and  cc-  in  (3),  we  have  the  equation  in  terms  of  the  semi- 
diameters,  a'^  f  +  h'^  x^  =  c^' b'%  (4) 

which  is  of  the  same  form  as  the  equation  of  the  ellipse  referred 
to  its  axes,  the  semi-diameters  having  replaced  the  semi-axes. 

CoR.  1.  The  equation  of  the  hyperbola  referred  to  conjugate 
diameters  is  ^,,^,  _  ^,o^,  ^  _  ^^„^,,^  ^^^ 

since  the  onh'  change  in  the  above  would  be  that  arising  from 
the  minus  sign  of  6'-'  in  the  equation  of  tbe  hyperbola. 

Cor.  2.  The  equations  of  the  tangents  to  the  ellipse  and 
hyperbola  referred  to  conjugate  diameters  are 

a'-yy'±b"xx'=±a"b'-,  (6) 

since  the  only  change  in  the  process  of  Art.  104  would  be  that 
arising  from  the  substitution  of  a'  and  b'  for  a  and  b. 

111.  The  squares  of  ordinates  j^aycdlel  to  any  diameter  of  an 
ellipse  are  to  each  other  as  the  rectangles  of  the  segments  into 
ivhich  they  divide  its  conjugate. 

Let  P'3I',  P"M",  be  the  ordinates  parallel  to  any  diameter 
BB',  and  meeting  its  conjugate  AA'  in  3/'  and  M".  Then, 
a'-y^-{-b''X^=  a''^b''^  being  the  equation  of  the  ellipse  referred  to 
these  diameters,  we  have  for  the  points  P'  and  P" 


152  ANALYTIC   GEOMETRY. 

a'-  a'^ 

Dividing,         ^^^—  = =  -^^ ■ '-^ '-. 

y'"      a"-x"'      {a'-\-x"){a'-x") 

or  P'i¥'2 :  P"M"' :  :  AM' .  M'A' :  AM"  .  M"A'. 


112.  The  squares  of  ordinates  2'>cirallel  to  any  diameter  of  an 
hyperbola  are  to  each  other  as  the  rectangles  of  the  distances  from 
the  feet  of  the  ordinates  to  the  vertices  of  the  conj\igate  diameter. 

113.  The  parameter  of  an  ellipse  is  a  third  p)roportional  to 
the  transverse  and  conjugate  axes. 

The  axes  being  conjugate  diameters,  Art.  Ill  applies,  and 

y'^ :  y'2 :  :  (a  +  x)  (a  -  x')  :  (a  +  x")  (a  -  x") . 

Let  P'  coincide  with  the  exti'emity  of  the  conjugate  axis,  and 
P"  with  that  of  the  parameter.     Then 

y'=b,   y"  =  py   x'=0,   x"=ae, 

and  the  proportion  becomes 

b- :  2^':  :  a' :  a' {I  -  e') . 

But       l-e-  =  -;  .-.a-ib-iib-ip-,     or     2a:  2b  : :  2b  :  2p. 
a- 

114.  Any  ordinate  to  the  transverse  axis  of  an  ellijjse  is  to  the 
corresponding  ordinate  of  the  circumscribed  circle  as  the  conjugate 
axis  of  the  ellipse  is  to  its  transverse  axis. 

From  the  equation  of  the  ellipse  ?/-  =  —  (a^— a^),  and  that  of 

a- 

the  circumscribed   circle  y{'  =  cr—  x^,  where  y  and  y^  are  the 
ordinates  corresponding  to  the  same  abscissa  x^  we  have 

9  O 

115.  Tlie  sum  of  the  squares  of  conjugate  diameters  to  the 
ellipse  is  constant  and  equal  to  the  sum  of  the  squares  of  the  axes. 


OBLIQUE   AXES. 


153 


Let  x',  y\  be  the  coordinates  of  A'  (Fig.  76),  and  a;",  ?/", 
those  of  B'.     Since  B"B'  is  parallel  to  the  tangent  at  A',  its 

equation  is  ?/  = — x.    Combining  this  with  ci?y-+  b-xr  =  a-b-, 

a-y' 

to  determine  the  intersections  B'  and  B",  we  find 


ay' 


bx' 


x=:x"=±-^,      and     y  =  y"=:f—-. 
b  '       '  a 


But 


a~  a- 


and       6'2  =  x"'+  y"'-  =  ^  +  ^  =  "' 
^  b-  a-        b- 


0       /       O  19  \ 


I   b'   ,9 


9      ,        0  Co  fO  9  9         19 

=  o-H ^ — X-  =  cr—  e-.i;-. 


Hence 


ce^j^h'-  =  a-+b-. 


116.  r/ie  difference  of  the  sqtiares  of  conjugate  diameters  to 
the  hyperbola  is  constant  and  equal  to  the  difference  of  the  squares 
of  the  axes. 

Let  a;',  y,  be  the  coordinates  of  A\  and  x'\  y",  those  of  B'. 


Fig.  77. 


154  ANALYTIC    GEOMETEY. 

The  equation  of  B"B'  is  ?/ = x.     Combining  this  with  the 

a'y' 

equation  of  the  y-hyperbola,  c^y"-—  h-a?  =  0.-11^,  to  determine  the 

intersections  B'  and  B",  we  find 

„      ,  ay'             „      ,  bx' 
x  =  X  =  ±  -^,    y=zy"=± 

b  a 

Hence 

a"=  x''+  2/"=  ^"+  -  (x"-  a')  =  ^-'-^x"--  b"=  e'x"--  b' ; 


'b\^,2_ 


a^ 


and      6'2  =  a;"2+2/"^  =  ^"  +  ^  =  ^, 

0"  a-        Ij- 

Hence  a''- -  V' =  a' -  h\ 


b\,n 


+  —x' 
a' 


117.  The  rectangle  of  the  focal  radii  draicn  to  the  extremities 
of  any  diameter  of  an  ellipse  is  equal  to  the  square  of  the  semi- 
conjugate  diameter. 

Let  {x',  y')  be  one  extremitj'  of  the  diameter.  Then,  if 
r  and  r'  represent  the  focal  radii,  rr' —  {a  —  ex')  {a-\- ex') 
(Art.  57).  Let  (x",  y")  be  the  extremity  of  the  conjugate 
diameter  whose  length  is  2b'.     Then 

„2,/2  f.2^1-2 

b'-^=:x"-'^y"-^  =  ^  +  ^-  (Art.  115) 
b'  a'' 

=  ^.^  (a2_  cc'2)+  ^  =  «2_  ^^^co"  =  a--  e'x'\ 
This  property  is  also  true  of  the  hyperbola. 

118.  The  area  of  the  parallelogram  formed  by  tangents  at  the 
extremities  of  conjugate  diameters  to  the  ellipse  and  the  hyperbola 
is  constant,  and  equal  to  the  area  of  the  rectangle  on  the  axes. 

Draw  OD  perpendicular  to  one  side  of  the  parallelogram 
(Figs.  76,  77).     Tlien  the  area  of  the  parallelogram  is 

4  OB'.  OA' .  sin  B'OA'=  4  OB'.  OA' .  sin  OA'T=A  OB'.  OD. 


OBLIQUE   AXES.  155 

The  normal  form  of  the  equation  of  the  tangent  at  A'  is 

a^yy'  ±  b'-xx'  g:  a'-'  b'-  _  j. 

■Va*y"-j-]^x"      ~    '  ^ 

Hence  the  distance  from  the  origin  to  the  tangent  is 

f=  =  T7  (Ai'ts-  115,  116), 


\    6^    "^    ce 

and  \OB.OD  =  \y-^^' =  ^ah. 

h' 

119.    To  find  the  equal  conjiKjate  diameters  of  the  ellipse. 
Equating  the  vahies  of  a'~  and  b''^  (Art.  110), 


a^  sin"y  +  6"  cos-y  a-  sin-yi  +  Z/-  cos^y^' 

whence         a?  sin^yi  +  b'  cos-yi  =  or  sin-y  +  W  cos^y  ; 
or,  transposing, 

a-(sin-yi  —  sin-y)  =  Zy-(cos^y  —  cos-yi)  =6-(sin-yi  —  sin^y), 

since  cos-^  =  1  —  siu-yl. 

Hence  (a^- 6^)  (sin- yi  -  sin-y)  =  0,  (1) 

and  therefore  sin^yi  =  sin-y.     Since,  in  the  ellipse,  if  y  is  acute, 
yi  is  obtuse,  and  the  sines  are  equal, 

yj=180°  — y    and    tanyi  =  — tany. 

7  - 

Substituting  this  in   tauy  tanyi  =  —  —  5 

a- 

the  equation  of  condition  for  conjugate  diameters  to  the  ellipse, 

tany  =  ±-:    .•.    tanyi=  —  tany  =  q: -• 
a  a 

Hence,  rt7ien  the  diameters  are  equal,  the  angles  they  make 
toith  the  transverse  axis  are  supplementary  and  the  diameters  fall 
on  the  diagonals  of  the  rectangle  on.  the  axes. 


156  ANALYTIC    GEOMETKY. 

Cor.  1.  If  o  =  b,  (1)  is  satisfied  iDdependently  of  y  and  yi; 
or,  in  the  circle  every  diameter  equals  its  conjugate. 

Cor.  2.     For  the  hyperbola,  (1)  becomes 

(a-  +b-)  (sin-yi  —  siiry)  =  0, 

which  cannot  be  satisfied  for  sin^yi  =  sin-y,  since  in  the  hyper- 
bola both  angles  are  acute  and  this  condition  would  make  them 
coincide.  Hence  the  hyperbola  has  no  equal  conjugate  diameters. 
From  a.'-  —  b'-  =  a-  —  b',  however,  we  see  that  if  a  =  &,  then 
a'  =  b' ;  or,  every  diameter  in  the  equilateral  hyperbola  equals  its 
conjugate. 

SUPPLEMENTAL    CHORDS. 

*  120.  Defs.  Straiglit  Hues  draw^u  from  auy  point  of  an  ellipse 
or  an  hyperbola  to  the  extremities  of  a  diameter  are  called  sup- 
plemental chords. 

Tjuis,  /S"Q,  QS'  (Figs.  76,  77)  are  supplemental  chords. 

121.  If  a  chord  of  an  ellipse  or  hyperbola  is  parallel  to  a  diam- 
eter, the  suirplementcd  chord  is  p^araUel  to  the  conjugate  diameter. 

Let  A" A'  (Figs.  76,  77)  be  a  diameter,  and  S"Q  the  parallel 
chord.  Draw  the  supplemental  chord  QS',  and  let  x\  y\  be  the 
coordinates  of  !S\  and  therefore  —  x',  —  y\  those  of  S".  The 
equation  of  S"Q  will  be  y  +  ?/'  =  a"{x  +  x')  (Art.  31) ,  and  that 
of  S'Q,  y  —  y'  —  a'(x  —  x') .  Combining  these  equations  by  mul- 
tiplication, y'^—  y'~  =  a'a"(x^  —  x'-) ,  in  which  x  and  y  are  the  co- 
ordinates of  Q  (Art.  36).  But  S'  and  Q  are  on  the  curve  ;  hence 
a-y''^±  b-x'"^  =  ±  d'b-  and  ahf±  b-.n?  =  ±  a^b- ;  or,  by  subtraction, 

y^  —  y'^  =  ^:  —  (a;'-  —  x'') .     f^quatiug  these  two  values  of  y-—  y'-, 

we  have  a'a"  =  0—-     Rut  this  is  the  condition  for  conjugate 

a-  J  2 

diameters,  viz. :  tan  y  tan  y,  =  q:  —  (Arts.  90,    93).      Hence   if 

a- 

d=  tany,  a"=  tany,,  and  conversely. 


OBLIQUE   AXES.  157 

Cor.  1 .  To  drato  a  tangent  at  a  given  point  of  the  curve,  as 
-4',  draw  the  diameter  A' A"  and  any  parallel  chord  as  S"Q. 
Draw  the  chord  QS'  supplemental  to  S"Q.  A  line  parallel  to 
QS'  through  A'  is  the  required  tangent. 

Cor.  2.  To  draw  a  tangent  X)arallel  to  a  given  line,  as  MN, 
draw  any  chord  QS'  parallel  to  it,  and  the  supplemental  chord 
QS".  Then  the  diameter  A" A',  parallel  to  S"Q,  determines  the 
points  of  tangenc}'  A"  aud  xV. 


PARABOLA   REFERRED    TO   OBLIQUE    AXES. 

122.   Equation  of  the  parabola  referred  to  any  diameter  and 
the  tangent  at  its  vertex. 

The  formulffi  for  transforming  from  rectangular  to  oblique 
axes,  the  new  origin  being  at  0',  are  (Art.  22,  Eq.  3) 

x  =  Xo-{-XiCosyj-yiCOSyi,     2/  =  2/o  +  ^isiny  +  ?/isinyi.    (1) 

But  y  =  0,  since  the  new  axis  of  X  is  parallel  to  the  primitive 

one,  hence  cosy  =  1,  siuy  =  0.     Also  tany,  =  —•>    since  the  new 

?h 
axis  of  Fis  tangent  to  the  curve  at  (x^,  y^)  (Art.  104,  Ex.  4)  ; 
hence  from 


smy,  siny, 

tanyi  = ^  =  —        ' 

cosyi      Vl-sin-yi 


we  have  sin  v,  = ^ 


sin  yi  = 


Vyo'+P' 


and  therefore         cos  y^  =  Vl  —  sin- y  =  —  •'^° 
Substituting  these  values  in  (1),  they  become 

x-  =  a-o  +  x,+  — M^,     y  =  yo+      ^'^ 


Substituting  these  values  in  the  equation  to  be  transformed, 
y^  =  2px,   and  remembering  that,  since    0'  is   on  the  curve, 


158 


ANALYTIC   GEOMETRY. 


t/q-  =  2pxo,  we  have,  after  omitting  the  subscripts  of  x  and  y, 

f^-'jy^'+p'U,  (2) 

which  is  the  required  equation. 

Cor.  1 .     r/o  =  MO'  =  MN  tan  MNO'  =  p  cot yi,  O'X  being  the 
normal  and  MN=  subnormal  =p.     Hence 

^M±P^=2p{l  +  cot^yj)  =2pcosec2yi.--^^. 

and  (2)  ma}'  be  written 


sm' 


„2_    2p 


V  =  -i~^, —  X. 
sm-  Yi 


71 


(3) 


Fig.  78. 

CoR.  2.     From  the  polar  equation  of  the  parabola, 

P 


r  = 


1  —  cos  6 


making  6  =  yi,  we  have 


r  =  FQ  = 


P 


1  —  cosyi 


making  ^=  180° +  yi, 


r=FQ'  = 


P 

__ —     .— .  ■■■« 

1  +  cos  yi 


OBLIQUE   AXES.  159 

Hence  QQ'  =  FQ  +  FQ'  =  4|-  ; 

sin-yi 

or,  representing  QQ'  by  2j>',  (3)  may  be  written 

/  =  •2p'x.  (4) 

Thus  the  equation  of  the  parabola  referred  to  any  diameter 
and  the  tangent  at  its  vertex  is  'if  =  2p'x,  2p'  being  tlie  focal 
chord  parallel  to  the  tangent,  and  becoming  2p  when  the  diam- 
eter is  the  axis. 

CoR.  3.  The  equation  of  the  tangent  referred  to  a  diameter 
and  the  tangent  at  its  vertex  is  yy'=p\x  +  x'),  since  the  only 
change  in  the  process  of  Art.  104  is  that  arising  from  the  sub- 
stitution oi  p'  for  p. 

123.  The  squares  of  orclinates  to  any  dinmeter  of  a  parabola 
are  to  each  other  as  their  corresponding  abscissas. 

Referred  to  any  diameter  and  the  tangent  at  its  vertex  the 
equation  of  the  parabola  is  y^  =  2p'  x.     Hence  for  the  points 

pi  gnri    P" 

^  ana  i-  ,  ^,o  ^  ^^,^,^    ^„,  ^  ,^^y^„  . 

or,  by  division,  ^2=^' 


ASYMPTOTES. 

r    124.   Equation  of  the  hyperbola  referred  to  its  asymptotes. 

The  asymptotes  being  oblique  except  when  the  hyperbola  is 
rectangular  (Art.  92),  we  use  the  formulae  for  passing  to 
oblique  axes  with  the  same  origin, 

X  —  x'l  cos y  +  ?/i  cos 7i,     y  =  x^  siny  +  y,  sin yj ; 

and,  since  the  asymptotes  coincide  with  the  diagonals  of  the 
rectangle  on  the  axes, 

—  h         .  h  a 

smy= —  ,   smyi= —  ,   cosy  =  cos yi  = 


Va^  +  6^  Va'  +  62  ^a'  +  b^ 


160 


ANALYTIC   GEOMETRY. 


The  formulae  therefore  become 
a 


x  = 


Va-+  h' 


(a?i+yi),    y 


iVi-Xi). 


Substituting  these  values  in  a-y^—b-xr  =  — irb^,  and  omitting 
the  subscripts,  we  obtain 


xy  = 


Fig.  79. 

Hence   the   general   form  of    the   equation  of    the   hyperbola 
referred  to  its  asymptotes  is    xy  =  m,    in  which  m  is  constant. 

CoR.   1.     The  equation  of   the   ^'-hyperbola  referred  to  the 

a-  +  0- 


same  axes  is  xy  =  — 


(Art.  71). 


Cor.  2.     The  equation 

Bxy  +  Dy  ■j-Ex  +  F=0 

is  the  general  equation  of  the  hyjm-bola  referred  to  axes  parallel 
to  its  asymptotes.  For,  transforming  xy  =  m  to  parallel  axes 
by  the  formuhie  x^Xq+x^,  y  =  yo+yi,  we  have,  after  dropping 
the  subscripts,  07/ +  .To?/ 4-.'/o-^' +  -^"(,?/o  =  0,  which  is  the  above 
form.     The  equations  of  the  asymptotes  are  evidently  ar  =  —  2/0, 


c) 


OBLIQUE   AXES. 


161 


125.    The  intercepts  of  the  secant  between  the  hyperbola  and  its 
asymptotes  are  equal. 

Let  P',  P"  (Fig.  79),  be  any  two  points  of  the  hyperbola, 

x'—x" 
the  equation  of  the  secant  P'P".     Making  a;  =  0  in  this  equa- 
tion, 

y-y'=D'Q'=f''^-y^f. 
x'—  x" 

But  y'x'=y"x"=  m,  since  tlie  points  are  on  the  curve. 


Hence- 


D'Q'= 


_y"x- 


V   X 


x'-x" 


■■y"=P"M". 


Hence  the  triangles  P"M"Q",  Q'D'P,  being  equiangular,  and 
having  a  side  in  one  equal  to  a  side  in  the  other,  are  equal, 
and  P"Q"=P'Q'. 

Cor.  To  construct  the  Jiyperhola  when  the  axes  are  given: 
draw  the  asymptotes,  the 
diagonals  of  the  rectangle  on 
the  given  axes,  and  through 
the  extremities  of  the  trans- 
verse axis,  as  A,  draw  11', 
22',  33',  etc.,  and  make 
IP',  2P",  3P"',  etc.,  equal 
respectively  to  ^1',  J. 2',  ^3',  0 
etc.  Then  P',  P",  P'",  etc., 
are  points  of  the  curve.  By 
a  similar  method  we  may 
construct  the  curve  when  the 
asymptotes  and  one  point  of 
the  curve  are  given.  Fig.  go. 


126.  The  area,  of  the  triangle  formed  by  any  tangent  ivith  the 
asymptotes  is  constant,  and  the  tangent  is  bisected  at  the  point  of 
contact. 


1B2  AJSTALYTIC    GEOMETRY. 


The  equation  of  the  secant  P'P"  (Fig.  79)  is 

From  the  equation  of  the  curve, 

x"u" 
x^y'z=  x"y"=  m,  .-.  y'=  " — ^• 

x' 

y'—  v" 

The  fraction —  therefore  becomes 

X  —x" 


x"y" 
x' 

-y" 

= 

y". 

X'' 

x'- 

■x" 

or,  when  P"  coincides  with  P',  —  -^  .     Hence  the  equation  of 
the  tangent  TT'  is  ^' 

v' 

y-y'=-~{x-x'), 

X 

and   its  intercepts  are   y  =  0T=2y',  x=  0T'=  2x'.     Hence 
P'  is  the  middle  point  of  TT'  (Art.  6). 
Again,  the  area  of  the  triangle  OTT'  is 

2  2 

=  2  a;'?/' 2  sin  TO  A  cos  TO  A  =  4x'y'    ,       —     ,  =  «&, 

since  x'y'= — — —     Hence  the  area  of  the  triangle  is  constant 
and  equal  to  the  rectangle  on  the  semi-axes. 

Cor.  To  construct  a  tangent  at  any  point,  as  P',  when  the 
asymptotes  are  given,  draw  the  ordinate  P'3I'  and  make 
M'T'  =  OM'.     P'T'  is  the  tangent. 

127.  Tangents  at  the  extremities  of  conjugate  diameters  meet 
on  the  asymptotes. 

The  equation  of   the  straight   line  P'B'  (Fig.   70),  the  co- 


OBLIQUE   AXES.  163 

ordinates  of  P'  being  x',  y',  and  those  of  B'  being  -^,    — • 
(Art.  IIG),  is  0       a 

,      bx' 

y 

y-y= — w(^-^)' 
& 

or  y-y'=-~{x-x'). 

a 

But  the  equation  of  OT"  is  y  = x',  hence  P'B'  is  parallel 

a 

to  the  as3-mptote  OT'.     Again,  the  middle  point  of  P'B'  is 

i(x'-\-^\    ify+^'M      —     f^^'+(^y'     bx'+ay' 


°'  '~^6~'  ^^r~" 


which   satisfy  y  =  -  x.     Hence  the  straight  line  joining  the  ex- 
a 

tremities  of  conjugate  diameters  is  parallel  to  one  asymptote  and 
bisected  by  the  other.  But  the  diagonals  of  a  parallelogram 
bisect  each  other,  and  P'B'  is  one  diagonal  of  a  parallelogram 
of  which  OP'  and  OB'  are  adjacent  sides  ;  hence  the  other 
diagonal  coincides  with  the  asymptote,  or  the  tangents  at  P' 
and  B'  meet  on  the  asymptote. 


CHAPTER   IV. 
LOCI. 


-o-Oj^^OO- 


128.  Classification  of  loci. 

Wlieu  the  relation  between  x  and  y  can  be  expressed  by  the 
six  ordinary  operations  of  algebra,  viz.,  addition,  subtraction, 
multiplication,  division,  involution,  and  evolution,  the  powers 
and  roots  in  the  latter  cases  being  denoted  by  constant  exponents, 
the  function  is  called  an  algebraic  function  ;  and  loci  whose 
equations  contain  only  algebraic  functions  are  called  algebraic 
loci. 

Algebraic  loci  are  classified  according  to  the  degree  of  their 
equations  as  loci  of  the  first,  second,  etc.,  orders.  We  have 
seen  that  there  is  but  one  locus  of  tlie  first  order  ;  that  is,  whose 
equation  is  of  the  first  degree,  nameh',  the  straight  line  ;  and 
that  all  loci  of  the  second  order  are  conies.  All  loci  whose 
equations  are  above  the  second  degree  are  called  higher  plane 
curves. 

A  function  whicli  involves  a  logarithm,  as  a;  =  logy,  is  called 
a  logarithmic  function ;  one  in  which  the  variable  enters  as  an 
exponent,  as  y  =  iij',  an  exponential  function.  If  a  is  the  base 
of  the  logarithmic  system,  the  latter  function  is  evidently 
another  way  of  expressing  the  former.  Functions  involving 
the  trigonometrical  elements,  as  y  =  smx,  .«=sin~^?/,  etc., 
are  called  circular  functions.  y  =  smx  and  .T=sin"'y  ^^re 
different  forms  of  the  same  relation,  the  former  being  called  the 
direct,  and  the  latter  the  inverse  circular  function.  It  may  be 
shown  that  logarithmic,  exponential  and  circular  functions  can- 
not be  expressed  by  a  finite  number  of  algebraic  functions,  and 


LOCI.  165 

for  this  reason  they  are  called  transcendental  functions.  A 
transcendental  equation  is  one  involving  transcendental  func- 
tions, and  the  locus  of  such  an  equation  is  called  a  transcen- 
dental curve. 

The  exercises  which  follow  will  afford  the  student  practice 
in  the  production  of  the  equation  of  a  locus  from  its  definition. 
In  all  cases  the  object  is  to  find  a  relation  between  the  given 
constants,  x,  and  y ;  the  latter  being  the  coordinates  of  any 
point  of  the  required  locus.  Any  such  relation,  when  stated  in 
the  form  of  an  equation,  will  be  the  equation  sought,  whatever 
the  axes  ;  but  the  simplicity'  of  both  the  solution  and  the  result- 
ing equation  will  depend  upon  the  choice  of  the  axes.  The 
student  will  observe  two  cases  :  first,  when  the  given  conditions 
furnish  directly  a  relation  between  x  and  y ;  second,  when  the 
conditions  involve  other  variables  ;  and  in  tliis  case  these  con- 
ditions must  afford  a  sufficient  number  of  independent  equations 
to  permit  the  elimination  of  all  the  variables  except  x  and  y. 
Thus,  if  71  variables  ai'c  involved  exclusive  of  x  and  y,  the 
conditions  must  furnish  w  +  1  equations. 


166 


ANALYTIC    GEOMETRY. 


SECTION   XI.  — LOCI   OF   THE   FIRST    AND    SECOND 

ORDER. 


129.     1.   Given  the   base  of  a  triangle  and  the  difference  of 
the  squares  of  its  sides,  to  find  the  locus  of  the  vertex. 

Let  h  be  the  given  base  and  d^  the  constant  difiference.     Take 

the  base  for  the  axis  of  X,  and  its  left- 
hand  extrerait}'  for  the  origin,  x  and  y 
being  the  coordinates  of  the  vertex. 
Then,  by  condition,  OP-  —  BP-=d-,  or 

x-  +  f-[{b-xY  +  f]=d\ 
dr  +  6- 


whence 


X: 


M  B 


2b 


Fig.  81. 


to  Y,  at  a  distance  from  it  equal  to 

b 


Hence  the  locus  is  a  straight  line  parallel 

d'  +  b' 


2b 


If  the  triangle  is 


isosceles,  d  =  0,  and  x  =  "  •    In  this  ease  the  conditions  furnish 

2 

directly  the  relation  between  x  and  y. 

2.  To  find  the  locus  of  the  middle  point  of  a  rectangle  inscribed 
in  a  given  triangle. 

Let  a  =  altitude  of  the  triangle,  b  and  c  the  segments  of  the 
base,  the  axes  being  taken  as  in  the  figure.  Then  the  equations 
of  AB  and  AC  are  known  ;  namely, 


X 


V 


X 


-  +  •^=1,  and  -  +  -'  =  1 


.V_ 


a 


Now  the  abscissa  of  P  is  the  half  sum  of  the  abscissas  of  Q  and 
R ;  and  if  y  —  k  be  the  altitude  of  the  rectangle,  and  this 
value  be  substituted  for  y  in  the  above  equations,  we  find 


x^  = 


a  —  k 


a 


b, 


a  —  k 


a 


LOCI   OF   THE   FIRST   AND    SECOND   ORDER. 


167 


Hence  x  =  abscissa  of 


2 


a 


But  the  ordinate  of  P  =  ?/  =    • 

2 

This  condition  enables  us  to 
eliminate  the  variable  A:  from  the 
above  value  of  x ;  substituting 
therefore  A;  =  2y,  we  have 

2ax  =  {a  -  2>j)  {b  +  c) , 

a  straight  line  bisecting  the  base  and  altitude,  since  its  intercepts 
are  ^  (b  -\-c)  and  4-  a. 

3.  To  find  the  locus  of  a  point  so  moving  that  the  square  of 
its  distance  from  a  fixed  lioint  is  in  a  constant  ratio  to  its  dis- 
tance from  a  fixed  line.         ^ 

Let  B  be  the  fixed  point,  ^X  the  fixed  line  and  axis  of  X, 
the  axis  of  F  passing  through  5,  and  0B  =  a.     Then,  m  being 

BP- 

the   constant    ratio, =  m.     But  BP- =  x- -{-(y  —  a)-,   and 

PM=  y.     Hence 

2/2  +  ar'_(2a  +  m)^  +  a2  =  0, 
which  is  the  equation  of  a  circle  whose 


centre   is   at  f  0 


2  a  +  m 


and   whose 


radius   is  |^  V4a?» +  m^  (Art.  50).     If 
the  point  is  on  the  line,  a  =  0,  the  cen- 
tre is  f  0,  —  1  and  the  radius  =  — 
'27  2 


4.  The  squares  of  the  distances  of  a  jjoint  from  two  fixed 
points  are  as  m  to  n.     Find  the  locus  of  the  point. 

Let  (a,  0),  (0,  &),  be  the  coordinates  of  the  fixed  points,  the 
axes  being  assumed  to  pass  through  them,  and  P  any  point  of 
the  locus.     Then,  A  and  B  being  the  fixed  points, 


168  ANALYTIC    GEOMETRY. 

PB"  _  x-  +  {y-h)-  _  m  . 
PA'~  xj-  +  {x-ay~n  ' 

or,  clearing  of  fractions  and  reducing, 

o  ,     <>       2  nb        ,    2  am       ,  nh-  —  ma^      ^ 
y-  +  x- y  H X  + =  0, 

11  —  7)1         n  —  m  n  —  m 

a  circle  whose  centre  is  ( -»    — — ),  and  radius  is 

\     n  —  III     11  —  iiij 

1 


■■y/mnia^  +  b-)  (Art.  50). 

If  6  =  0,  or  a  =  0,  that  is,  if  both  the  points  are  on  the  same 
axis,  the  centre  is  on  that  axis.  If  a  =  b  =  0,  the  centre  is  at 
the  origin  and  i2  =  0,  or  the  locus  is  a  point ;  unless  also  m  =  ?i, 

when  B  =  — 
0 

5.  Find  the  locus  of  the  vertex  of  a  triangle  having  given  the 
base  and  the  sum  of  the  squares  of  the  sides. 

Ans.  A  circle  whose  centre  is  the  middle  x>oint  of  the  base. 

6.  Given  the  base  of  a  triangle  and  the  ratio  of  its  sides,  find 

the  locus  of  the  vertex. 

OP 

Let   6  =  base   of  the  triangle  (Fig.    81),   and —  =  m,    the 

ratio.     Then  OP' =  m2P5^  or  ^^ 

x'^^f=i,r{y\+{b-xy), 

whence  ?/  +  ^  +  z z^—  z ;  =  0  5 

a  circle  whose  centre  is  f -?    0  ),  and  radius  is 


\  —  111?      j  1  —  m' 

7.  From  one  extremity  A'  of  a  diameter  AA'  to  a  circle  a 
secant  is  drawn  meeting  the  circle  at  P.  At  P'  a  tangent  to  the 
circle  is  drawn,  and  from  A  a  perpendicular  to  this  tangent.  T7ie 
perpendicidar  produced  meets  the  secant  at  P.  Find  the  locus 
of  P. 


LOCI   OF   THE   FIRST    AND    SECOND   ORDER.  169 

Let  the  diameter  be  the  axis  of  X  and  the  centre  the  origin. 
Let  {x',  y')  be  the  coordinates  of  P' ;  then  (Art.  32) 

is  the  eqnation  of  the  secant ;  the  equation  of  the  tangent  at 
P'  is  yy'  +  xx'  =  jB^,  hence  the  perpendicular  on  the  tangent 
from  A  is  , 

y  =  l,{x-R).  (2) 

Combining  (1)  and  (2),  to  find  P,  we  have 

x=2x'  +  R,     y  =  'ly\  (3) 

But  (x',  ?/')  is  on  the  circle,  hence  x'-  +  ?/'-  =  Bj^.  Substituting 
in  this  equation  the  values  of  xJ  aud  ?/'  from  (3) ,  we  have 

a  circle  whose  centre  is  at  (i?,  0),  that  is,  at  A^  and  whose 
radius  is  '2R  =  AA . 

8.  A  line  is  draion  parallel  to  the  base  of  a  triangle,  and  the 
points  where  it  meets  the  sides  are  joined  transversely  to  the 
extremities  of  the  base;  find  the  locus  of  their  intersection.  Take 
the  sides  as  axes. 

Ans.  A  straight  line  through  the  middle  point  of  the  base  and 
the  opposite  vertex. 

9.  Given  the  base  and  sum  of  the  sides  of  a  triangle,  if  the 
perpendictilar  be  produced  beyond  the  vertex  until  its  whole  length 
is  equal  to  one  of  the  sides,  to  find  the  locus  of  the  extremity  of 
the  perpendictdar.  Ans.  A  straight  line, 

10.  Given  any  parallelogram,  ^    Q 
and  PP',  QQ',  lines  parallel  to 
adjacent  sides.      Prove  that  the 
locus  of  the  intersection   of  PQ      p/ 
and  P'Q'   is  a  diagonal  of  the    ^    qi     ^         ii 
parallelogram  (Fig.  84) .  Fig.  84. 


170 


ANALYTIC    GEOMETRY. 


11.  In  Fig.  84,  find  the  locus  of  the  intersection  of  BL  and 
PA^  A  and  B  being  fixed  points,  and  P  and  L  subject  to  the  con- 
dition that  0L  +  OB=OP+  OA. 

12.  A  line  cuts  two  fixed  intersecting  lines  so  that  the  area  of 
the  intercepted  triangle  is  constant.  Find  the  locus  of  the  middle 
point  of  the  line  (see  Art.  126). 

Let  OX,  OT,  be  the  fixed  hues  and  axes,  AOB  the  inter- 
cepted triangle,  m  the  constant  area,  <^  the  constant  angle  BOA, 
and  P  the  middle  point  of  AB.  Then  OM=x,  3IP=y,  and, 
since  P  is  the  middle  point  of  AB,  0A  =  2yani\  0B=2x. 
Hence 


area  BOA 


or 


xy. 


2x.2y.smcf) 

in  =  ^;r J 


m 


2sin^ 
an  hyperbola  whose  asymptotes 
are  the  fixed  lines  (Art.  124). 

13.  Given  ttco  intersecting 
fixed  lines  and  a  fixed  x>oint. 
A  line  is  drawn  through  the  fixed  point.  Find  the  locus  of  the 
middle  point  of  the  segment  iyUercejyted  by  the  given  lines. 

Let  OX,  OF  (Fig.  85),  be  the  fixed  lines  and  axes,  Q  the 
fixed  point,  its  coordinates  OP  =  m,  PQ  =  n,  AB  the  line,  and 
/-•  its  middle  point.     Then,  from  similar  triangles, 

0A{=2y)  :0B{=2x)  ::PQ{=n)  :PB(=2x-m), 

or  2xy  —  my  —  nx  =  0  ;  an  hyperbola  passing  through  Q,  whose 
asymptotes  are  x  =  —i   2/  =  -(Art.  124,  Cor.  2). 

14.  From  a  fixed  point  A  (Fig.  85),  a  line  AB  is  draivn  to 
meet  a  fixed  line  OX.  From  the  intersection  B,  a  constant  dis- 
tance BR  =  h  is  laid  off,  and  from  R  a  line  PQ  is  drawn,  mak- 
ing a  constant  angle  ivith  OX,  to  meet  AB  in  Q.  Find  the 
locus  of  Q. 


LOCI   OF   THE   FIRST    AND    SECOND   ORDER.  171 

Since  the  angle  BRQ  is  constant,  take  a  parallel  to  QR 
through  A  for  the  axis  of  Y,  and  the  fixed  line  OX  for  the 
axis  of  X,  and  let  OA  =  a.  Then,  OA  .  JiQ  :  :  OB  :  MB,  or 
a:  y  :  :  x-\-b:b  ;  whence  xy  -{-by  —  ab  =  0,  an  hyperbola  through 
A,  one  of  whose  asymptotes  is  the  fixed  line  and  the  other 
x^-b  (Art.  124,  Cor.  2). 

15.  To  find  the  locus  of  the  intersection  of  a  perpendicidar 
from  the  focus  of  a  parabola  on  the  normal. 

The  equation  of  the  normal  is 
and  that  of  the  perpendicular  is 

y 


y=z-'[^-.) 


Combining  these  to  find  the  point  of  intersection,  we  find  it 

to  be 

p^4-  2x'y'--{-22yy'-  '2px'y'-\-2'ry' 

^=        2?/'^+ 2  2/       '     ^=    2.^2+ 2p2 

In  this  pi'oblem  the  conditions  introduce  the  auxiliary  vari- 
ables a;',  y',  the  coordinates  of  the  point  of  contact  from  which 
the  normal  is  drawn.  But  this  point  is  on  the  parabola;  hence 
we  have  the  additional  condition  y''-=  2px'.  Eliminating  y'  by 
means  of  this  equation,  we  have 

Finally,  combining  and  eliminating  a;',  we  have 

2      P         P" 

a  parabola  on  the  same  axis,  whose  vertex  is  at  (l^,  0  ),  and 
whose  parameter  =  \  that  of  the  given  parabola. 

16.  The  locus  of  the  intersection  of  the  perpendicular  from  the 
focus  of  a  parabola  upon  the  tangent  is  the  tangent  at  the  vertex. 


172  ANALYTIC   GEOMETRY. 

The  equation  of  the  tangent  is 

and  of  the  perpendicular  upon  the  tangent  through  the  focus, 

Combining  these  to  find  the  intersection,  we  obtain,  on 
eUmiuatiug  y,  x{p  -j-'2x')=0,  which,  since  x'  cannot  be  nega- 
tive, is  satisfied  only  for  a;=0;  that  is,  the  intersection  is 
always  on  Y,  which  is  the  tangent  at  the  vertex. 

How  does  this  property  enable  us  to  find  the  focus  when  the 
curve  and  axis  are  given  ? 

17.  Throvgh  any  fixed  point  cliords  are  draivn  to  a  parabola. 
Find  the  locus  of  the  intersections  of  the  tangents  to  the  parcdtola 
at  the  extremities  of  each  chord. 

Let  .Tj,  2/1,  be  the  coordinates  of  the  point  through  which  the 
chords  are  drawn,  and  suppose  the  tangents  at  the  extremities 
of  one  of  these  chords  to  meet  at  (/*,  k).  Then  the  equation 
of  the  chord  is  (Art.  106) 

yk^p{x  +  h). 

But  the  chord  passes  through  the  fixed  point  (.rj,  yi) ,  hence 
y^k  =  p  {Xi+  h).  Now  this  is  the  equation  of  a  straight  line,  in 
which  h  and  k  are  the  variables ;  therefore  the  locus  of  (/i,  k) 
is  a  straight  liue. 

If  the  fixed  point  is  the  focus,  a^i  =^,  ?/,  =  0,  and  the  equation 

becomes  h  =  —  --     Hence  the  locus  of  the  intersection  of  pairs  of 
tangents  drawn  at  the  extremities  of  focal  chords  is  the  directrix. 

18.  Through  any  fixed  point  chords  are  drawn  to  the  ellipse 
(or  hyperbola) ;  to  find  the  locus  of  the  intersection  of  the  tan- 
gents at  the  extremities  of  each  chord. 

I^et  a^i,  ?/,,  be  the  coordinates  of  the  point  through  which  the 
chords  are  drawn,   the  tangents  at  the  extremities  of  one  of 


LOCI   OF   THE   FIRST   AND   SECOND   ORDER.  173 


them  meetiug  at   {h,  k) .     Then  the  equation  of   this  chord  is 

a?yk  ±  b-xh  =  ±  (rl/. 


(Art.  106) 

V  ^  _.!!..7.      1      1,9..  7  I        ,27,2 


But  this  chord  passes  through  the  fixed  poiut  (x-j,  ?/i),  hence 
ahj-^k  ±  b-xji=  ±  crV^.  Now  this  is  the  equation  of  a  straight 
line  in  which  h  and  7i'  are  the  variables  ;  therefore  the  locus  is  a 
straight  line. 

If  the  fixed  point  is  the  focus,  x^=  ae,  yx=  0,  and  the  equa- 
tion becomes  /*  =  -•  Hence  the  tangents  at  the  extremities  of 
e 

focal  chords  to  the  ellipse  and  hyperbola  intersect  on  the  directrix. 

19.  The  locus  of  the  intersection  of  the  perpendicular  from  the 
focus  of  an  ellipse  upon  the  tangent  is  the  circle  described  on  the 
transverse  axis. 

In  this  problem  the  equation  of  the  tangent  in  terms  of  the 
slope  (Art.  105,  Ex.  12), 


y  =  mx  +  Va-»i-+  6", 

is  most  convenient.     The  perpendicular  upon  it  from  the  focus 

is  1,  . 

y  = {x-ae). 

'tn 


From  the  former,  y~mx=  ^a-m^-^b'-,  and  from  the  latter, 
my  +  tc  =  ae.     Squaring  and  adding, 

(v'+a;-)(l  +m')  =  b--\-m-a-  +  a'e-  =  b-+ma'+cr^-^—^=  a-{i+m'), 

a^ 

which  eliminates  m,  giving  y"-\-  x-=  a-. 

This  property  is  also  true  of  the  hj'perbola. 

20.  Find  the  locus  of  the  intersection  of  pairs  of  tangents  to  a 
parabola  ivhich  intercej^t  a  constant  length,  on  the  tangent  at  the 
vertex. 

The  equations  of  the  tangents  are 

yy'  =p{x  +  x'),  (1) 

yy"  =  p{x  +  x^').  (2) 


174  ANALYTIC   GEOMETRY. 

The  equation  of  the  locus  will  be  found  by  combining  these 
and  eliminating  x',  y\  x",  and  y".  To  effect  this  elimination  we 
have  the  equations  of  condition, 


y''  =  2px', 

(3) 

y"-'  =  22^x", 

(4) 

w'-w" 

— TT^-  =  a,  a  constant. 

(5) 

Substituting  in  (1)  and  (2)  the  values  of  x'  and  x"  from  (3) 
and  (4),  we  have 

yy'=2:)x  +  -^,  (6) 


,112 


yy"=px  +  ^-  (7) 

Substituting  from  (5)  y'=  2a-j-y"  in  (6)  and  combining  the 
result  with  (7),  we  have  y"—y  —  a\  which  substituted  in  (7) 
gives  y^=  2px  +  a?,  an  equal  parabola  with  the  same  axis,  and 

vertex  at  (  —  — -,  0 

21.  Parallel  chords,  as  QQ\  ivhose  centime  is  C,  are  draivn  to 
a  circle.  AA'  is  a  diameter  parallel  to  the  chords.  Find  the 
locus  of  the  intersection  of  AC  loith  the  radii  through  the  extremi- 
ties of  the  chords. 

Ans.  A  parabola  ivhose  axis  is  the  diameter  and  vertex  mid- 
loay  between  0,  the  centre.,  and  A. 

22.  Find  the  locus  of  the  intersection  of  tangents  drawn  at  the 
extremities  of  conjugate  diameters  of  an  ellipse. 

Ans.  2 aY-+  2  b'x^=  4 o'bK 

23.  Lines  RL,  R'L'  (Fig.  84),  are  draivn  parallel  to  one  side 
of  a  parallelogram  and  equidistant  from  the  centre.  Find  the 
locus  of  the  intersection  of  a  line  drawn  from  0  through  the 
extremity  R  of  one  of  the  parallels  ivith  the  other,  or  the  other 
produced.  Ans.  An  hyperbola. 


LOCI   OF   THE   FIRST   AND    SECOND   ORDER.  175 

24.  A  and  B  are  fixed  2wints.     Find  the  locus  of  P  when 

PD- 

=  o  constant^  D  being  the  foot  of  the  perpendicular 


ADxDB 

from  P  on  AB.  Ans.  An  ellipse. 

25.  To  find  the  locus  of  the  centres  of  all  circles  which  pass 
through  a  given  point  and  are  tangent  to  a  given  straight  line. 

Ans.  A  parabola. 

26.  If  a  variable  circle  touch  a  fixed  circle  and  a  fixed  straight 
line.,  the  locus  of  its  centre  is  a  parabola. 

27.  Given  the  base  of  a  triangle  and  the  product  of  the  tan- 
gents of  the  base  angles;  the  locus  of  the  vertex  is  an  ellipse. 

28.  The  base  and  area  of  a  triangle  is  constant;  the  locus  of 
the  vertex  is  a  straight  line. 


176 


ANALYTIC   GEOMETRY. 


SECTION  XII.— HIGHER    PLANE    LOCI. 


130.  The  limits  of  this  work  permit  a  reference  to  a  few 
ouly  of  the  higher  plane  curves  possessing  interesting  geometric 
properties. 

1.  The  cardioid.  Through  any  point  0  of  a  circle  a  secant 
is  drawn  cutting  the  circle  in  Q.  Reqnirecl  the  locus  of  a  point  P 
on  the  secant  tvhen  QP=  R,  the  radius  of  the  circle. 

Let  C  be  the  centre  of  the  circle,  CO  =  R  the  radius,  0  the 
pole,  and  OX,  a  tangent  at  0,  the  polar  axis.      Then 

OP=OQ+QP. 
But  OP=r, 

OQ  =  OD  cos  QOD  =  OD  sin  XOQ  =  2  i?  sin  0,  and  QP  =  r. 
Hence  r=^2Rsme  +  R.  (1) 

Discussion  of  the  equation.  For  6  =  0°,  r  =  R—  OA. 
As  6  increases,  r  increases,  and  when  ^  =  90°,  r='6R=  OB. 


Fig.  86. 


HIGHEE    PLANE   LOCI. 


177 


As  0  increases  from  90°  to  180°,  r  diminishes,  and  when  0  =■  180°, 
r=Ji=OE.  When  6  passes  180°,  sin  ^  becomes  negative, 
but  r  remains  positive  nntil  sin^  =  — ^,  when  ?•  =  0  ;  at  this 
point  ^=210°.  From  this  value  of  ^,  r  is  negative  and  the 
portion  OFC  is  traced,  r  being  —  7^  when  ^  =  270°.  From 
^=270°  to  ^  =  360°  the  portion  COA  is  traced,  r  becoming 
positive  again  when  0  =  330°. 

Rectangular  equation  of  the  cardioid.     Transferring  to 
the  axes  YOX  hy  the  formula  (Art.  24,  Eq.  4), 

r  =  Vic^+2/^5     sin^= — -^        , 

we  have  (ar'  +  y-  -  2  By) -'  =.  {x~  +  /)  E\ 

a  curve  of  the  fourth  order. 


(2) 


Trisection  of  the  angle.  The  cardioid  affords  a  metliod 
of  trisecting  an  angle,  as  follows.  Let  NCO  be  the  given  angle. 
With  the  vertex  C  as  a  centre  describe  any  circle,  and  construct 
the  cardioid  to  this  circle.  Only  that  portion  of  the  curve  in  the 
vicinity  of  NC  produced  need  be  constructed.  Produce  NC  to 
meet  the  cardioid  at  P,  and  draw  FO  and  QC.  Then  the  tri- 
angles CQP,  CQO,  are  isosceles  by  construction.      Hence 

NCO  =  COP  +  CPO=  CQO  +  CPO  =  QCP  +  2  CPO  =  3  CPO  ; 

or  CPO  =  lNCO. 

2.    The  conchoid.      Through   a  fixed  point  F  a   line   FP  is 


178  ANALYTIC  geo:metry. 

drawn  cutting  a  fixed  line  XX'  in  Q.     Required  the  locus  of  P 
when  QP  is  constant. 

Let  QP=  a,  FO  =  b,  the  distance  from  the  point  to  the  line, 
and  OX,  OY,  be  the  axes.  Draw  FS  and  PS  parallel  and  per- 
pendicular respectively  to  X'X.     Then  PM :  MQ  : :  PS  :  FS ; 

or  y  :  V«^  —  y-  : :  y  -\-b  :  x; 

whence  x-y~=  {y  +  h)-  ( «-  —  y-) ,  ( 1 ) 

a  curve  of  the  fourth  order. 

Discussion  of  tue  equation.  Solving  the  equation  for  x, 
we  have 


X  =  ±  -^— ' —  Va-  —  y- 

y 

The  curve  is  evidently  symmetrical  with  respect  to  Y.  When 
y  is  positive  and  equal  to  a,  .^•  =  0,  locating  the  point  A,  which 
is  a  limit  in  the  positive  direction  of  Y,  since  x  is  imaginary 
if  2/  >  a.  As  y  diminishes,  x  increases  numerically,  becoming 
±  00  when  2/ =0;  hence  the  curve  has  infinite  branches  in  the 
first  and  second  angles  with  X'X  for  their  common  asymptote. 
Since  x  is  real  for  negative  values  of  y  less  than  a  numerically, 
there  is  a  branch  in  the  third  and  fourth  angles.  When  y  =  —  a, 
or  — 6,  x=0,  locating  A'  and  F;  and  as  x  has  two  values 
numerically  equal  with  opposite  signs  for  values  of  y  between 
—  a  and  —  b,  the  locus  between  these  values  is  an  oval.  When 
y  is  negative  and  numerically  less  than  b,  x  increases  as  y 
diminishes  and  becomes  ±  co  for  y  =  0,  giving  infinite  branches 
with  the  axis  of  X  for  their  common  asymptote.  ?/  =  —  a 
evidently  limits  the  curve  in  the  negative  direction  of  Y. 

In  the  above  case  a>&.  If  a  =  ^,  i^  coincides  with  A'  and 
the  oval  disappears.  If  a  <  b,  all  negative  values  of  y  numer- 
ically greater  than  a  render  x  imaginary,  except  y  =  —  b.  wliich 
renders  x=0;  thus  the  oval  disappears,  but  .r  =  0,  y  =  —  b, 
satisfy  the  equation,  and  hence  must  be  considered  a  ]ioint  of 
the  curve.  Such  a  point  is  called  an  isolated,  or  conjugate 
point. 


HIGHER   PLANE   LOCI.  179 

Polar  equation  of  the  curve.  The  polar  equation  ma}- 
be  obtained  by  transformation,  or  directly  from  the  figure,  thus  : 
Let  F  be  the  pole  and  FS  the  polar  axis  ;  then 


r=FP  =FQ  +  QP=FO  amO  +  a, 
or  r=h  cosee  ^  +  a.  "  '  (2) 

Trisection  of  the  angle.  The  conchoid  also  affords  a 
method  of  trisecting  the  angle,  as  follows  :  Let  AFP  be  the 
given  angle.  Draw  any  line  OX  perpendicular  to  one  side, 
and  with  i*'  as  a  fixed  point,  OX  a  fixed  line,  and  PQ  =  2FQ, 
construct  the  arc  of  a  conchoid.  Only  that  portion  of  the 
curve  included  within  the  given  angle  need  be  drawn.  From 
Q  draw  QX  perpendicular  to  OX  and  join  its  intersection  with 
the  conchoid,  X,  with  F.  Bisect  QX  at  R,  draw  RL  parallel 
to  OX,  and  join  L  with  Q.  Then  the  triangles  LXQ^  LFQ, 
are  isosceles  ;  for,  since  QR  =^  RX, 

KL  =LX^LQ^^QP^FQ. 

Hence  the  angle 

AFX=FXq=LQX=^FLQ  =  ^LFQ,  or  AFX  ='^ AFP. 

Mechanical  construction.  The  conchoid  may  be  con- 
structed mechanically  as  follows :  Let  AA\  XX',  be  two  fixed 
rulers,  the  latter  having  a  groove  on  its  upper  surface.  Let 
FP  be  a  third  ruler,  having  a  peg  Q  fixed  on  its  under  side, 
which  is  also  grooved  to  slide  on  a  peg  at  F.  A  pencil  at  P 
will  trace  the  curve. 

3.  The  cissoid.  Pairs  of  equal  ordmates  are  drawn  to  the 
diameter  of  a  circle,  and  throngh  one  extremity  of  the  diameter  a 
line  is  draimi  through  the  intersection  of  one  of  the  ordinates  ivith 
the  circle.  Find  the  locus  of  the  intersection  of  this  line  with 
the  eqnal  ordinate  or  that  ordinate  jyrochiced. 

Let  OD  be  the  diameter  to  which  the  ordinates  are  drawn, 
and  the  axis  of  X,  the  tangent  to  the  circle  at  0  being  the 
axis  of  F.     Let  QM,  Q'M',  be  equal  ordinates.     Through  0 


180 


ANALYTIC   GEOMETRY. 


draw  OQ  (or  OQ')  ;  then  P'  (or  P)  is  a  point  of  the  locus. 
From  similar  triangles, 

0M:3fP:.0M':M'Q'; 

or,    R    being    the    radius   of    the 
circle. 


x:  y  ::  2R  —  x:  ^x  {'2  R  —  x), 
whence     y-  {2R  —  x)  =  x^.  (1) 

Discussion    of    the    equation. 
Solving  the  equation  for  y, 


y 


<x? 


\2R-x 


Fig.  88. 


which  shows  that  tlie  curve  is  sym- 
metrical with  respect  to  X.  If  x 
is  negative  or  greater  than  2R,yis 
imaginar}^ ;  hence  the  limits  along 
X  are  zero  and  2R.  As  x  in- 
creases, y  increases,  and  when 
x=2 R,  y=  ±  00  ;  hence  the  curve 

has  two  infinite  branches  which  have  the  tangent  at  D  for  a 

common  asymptote. 

Polar   equation  of  the  curve.      Let  0  be  the  pole  and 
OX  the  polar  axis.     Substituting  in  (1)  the  values 

x  =  rcos^,    ,?y  =  rsin^  (Art.  23,  Eq.  4), 

we  have  '/•^sin^^(2jK  — ?*cos^)  =  ^-^cos^^. 

But  sin- 0=1  —  cos^O  ; 

hence  2i?  -  2i?  cos'^  -  rcos^  =  0. 

1 


Substituting 


sec^ 


for  cos  6,  and  remembering  that 


sec- 6—  1  =  taii'-^, 
we  obtain  finally      r  =  2R  sin 6  tan 6. 


(2) 


HIGHER    PLANE   LOCI. 


181 


Duplication  of  the  cube.  This  curve  affords  a  method 
for  finding  the  edge  of  a  cube  whose  volume  shall  be  n  times 
that  of  a  given  cube,  as  follows :  C  beiug  the  centre  of  the 
circle,  take  CS  —  nCD,  and  draw  SD  intersecting  the  cissoid 
at  P.  Then  the  ordinate  P3f=  n  .  3fD,  and  the  cube  whose 
edge  is  P3f  is  n  times  the  cube  whose  edge  is  03f.  For,  P 
being  a  point  of  the  cissoid,  we  have  from  its  equation, 

03P       03P 


P3P-. 


MI) 


1         ' 
-  P3f 

n 


or     PM^=  n  .  OMK 


Let  c  be  the  edge  of  the  given  cube.     J'ind  o',  so  that 

c':c::  PM:  031,     or     c''^ :  c' :  :  P3P  :  03P. 

Then   c'^=  nc^,    for  P3P=  n  .  03P.      To  duplicate   the   cube, 
make  ?;.  =  2  ;  that  is,  take  CS  =  2  CD. 

4.  The  lemniscate.  To  Jind  the  locus  of  the  intersection  of 
the  perpendicular  from  the  centre  of  an  hyperbola  upon  the 
tangent. 


Assuming  the  form     y  =  mx  +  -Va-m-  —  b'^ 

of  the  equation  of  the  tangent  (Art.  105,  Ex.  12),  that  of  the 
perpendicular  is  , 


y  = 


X. 


m 


Substituting"  the  value  of  m  from  the  latter  in  the  former,  we 
have 


{f+x^y^arx^-Vf, 
a  curve  of  the  fourth  order. 

Polar  equation  of  the  cukvk. 
The  formulae  for  transformation 
being 

x--{-y-=f,  .i'=rcos^,  ?/=rsin^, 

we  have 

r^  =  a^  cos- ^  —  6^  sin-^, 
or     1^  =  a'-  —  (a-  +  b^)  sin' 0.    (2)  Fig.  89. 


(1) 


182 


ANALYTIC   GEOMETRY. 


DlSCCSSIOX    OF    THE    POLAR    EQUATION.         If    ^  =  0,     T  =  ±  tt, 

locatiug  A  and  ^i'.     As  $  increases,  r  diminishes  numerically 
a 


till  sin  (9  = 


;  that  is,  tiife*'  coim-ides-with  the  asym}itute, 


Vo-+  b' 
when  r=±0,  and  the  portions  ABO,  A'B'O^  are  traced,     r 

then  becomes  imaginar3%  and  remains  so   till   sin^  =  —   " 
again,  0  being  in  the  second  quadrant  [taii  7'  t- oingidiBg  with  th^ 


From  0  =  sin  ' 


a 


to  ^  =  180°,  sin^^ 

V  cr  +  0- 
is    diminishing    and    r   increasing    uumericall}',    the    portions 

OD'A',   ODA,  being  traced,  r  being   ±  a  when  $  =  180°. 

If  the  hyperbola  is  equilateral,  a  =  b,  and  (1)  becomes 

(;/-+ X-)- =  a- (.T- -?/-), 

and  (2)  in  like  manner  becomes 

r-  =  cr  (cos- ^  —  siu-^)  =  a-cos2^. 

In  tliis  case  r  is  imaginary  for  all  values  of  0  between  45° 
and  135°. 

5.    The  witch.      The  ordinate  to  the  diameter  of  a  circle  is 

prodxiced  till  its  entire  length  is  to  the  diameter  as  the  ordinate 

is  to  one  of  the  segments  ivith  which  it  divides  the  diameter^  these 

segments  being  taken  on  the  same  side.     Find  the  locals  of  the 

extremity  of  the  produced  ordinate. 

D 
Let  0A3  be  the  circle,  R  its  radius,  OD  the  diameter  and 

axis  of  Y,  0  the  origin,  and  the  tangent  at  0  the  axis  of  X. 


Fig.  90. 


HIGHER   PLANE   LOCI. 


183 


Then,  if  P  is  a  point  of  the  curve, 

PQ  :  DO  ■.:AQ:  QO, 


or  x:2  B  :  :  V2  Il>/  —  y- :  ?/, 

whence  off  =  4Ii\2  Ku  -f),  (1) 

a   loons  of   the    fourth    order.      Let   the    student   discuss   the 
equation. 

6.  To  find  the  locus  of  the  intersection  of  the  perpendicular 
from  the  vertex  of  a  parabola  upon  the  tangent. 

y-—2px  being  the  equation  of  the  parabola,  tliat  of  the  re- 
quired locus  is 


—  X' 


?r  = 


p 

2 


-|-  X 


P 


a  cissoid,  the  diameter  of  whose  circle  =j 

7.   Given  two  fixed  points  F  and  F',  to  find  the  locus  of  P  such 
that  PFxPF'=(^'y' 


Let  FF'  be  the  axis  of  X,  and  the  origin  in  the  middle  point 
oiFF'.  Then  {tf -\- x-y  =  2  c- {or  —  y-)  is  the  required  locus,  in 
which  c  =  ^FF'.    The  locus  is  the  lemniscate  (see  Ex.  4),  the 

hyperbola  being  rectangular,  and  c  = 


a 


8.  The  corner  of  a  rectangular  sheet 
of  paper  is  folded  over  so  that  the  sum 
of  the  folded  edges  is  constant.  Find 
the  locus  of  the  corner. 

By  condition.  OB  =  BP,  OA  =  AP, 
the  angle  at  P  is  a  right  angle,  and 
AP-}-  PB  =  a,  a  constant.     But 

AP''= AG'^  AE-  +  EP"-  =  {AO-yf+x\ 

.'.  3f+y^=2AO.y  =  2AP.y. 


184  ANALYTIC    GEOMETRY. 

Also  PB''=  OJBr-  =  PD'-+BD'=  f+  (x  -  0B)\ 

.-.  0^+  y^=  2  OB .  X  =  2 PB .  X. 

Substituting  the  values  of  AP  and  PB  from  these  equations  in 
AP-\-  PB  =  a,  we  have 

{x-+  ?/-)  {x  +  y)=  2  axy, 

a  locus  of  the  third  order. 


TKANSCENDENTAL   CUKVES. 


185 


SECTION   XIII.  — TRANSCENDENTAL  CURVES. 


131.   1.    The  logarithmic  curve. 

The  equation  of  this  curve  is  x  =  \ogy.  Assumiug  the  form 
y  =  a^,  in  which  a  is  the  base  of  the  logarithmic  system,  we 
observe  that  when  x=0,  ?/  =  l,  whatever  the  base;  hence  all 

/ 


Fig.  92. 

logarithmic  curves  cut  the  axis  of  F  at  a  distance  unity  from 
the  origin.  Again,  since  negative  numbers  have  no  logarithms, 
y  cannot  be  negative,  hence  these  curves  lie  wholly  above  X. 
If  X  is  positive  and  increasing,  y  increases,  but  more  rapidly 
than  X,  and  the  more  rapidly  as  the  base  is  greater ;  hence  the 
curve  departs  rapidly  from  X  in  the  first  angle,  and  the  more 

so  as  the  base  is  greater.     If  x  is  negative,  then  y  =  a~-'  =  — , 

from  which  we  see  that  as  x  increases  numerically,  y  decreases, 
and  the  more  rapidly  as  the  base  is  greater,  but  becomes  zero 
only  when  x  =  —  od  \  hence  the  curve  approaches  X  in  the 
second  angle,  and  that  axis  is  an  asymptote. 

2.  The  cycloid.  To  find  the  locus  of  a  point  in  the  circumfer' 
ence  of  a  circle  ivMch  rolls  without  sliding  along  a  fixed  straight 
line. 


186 


ANALYTIC    GEOMETRY. 


Let  OX  be  the  fixed  straight  line  and  axis  of  X,  0  the  initial 
position  of  the  generating  point  and  origin,  r  the  radins  of  the 
circle,  and  P  any  point  of  the  locus.     Then  OJr=  0N—3fN. 


Fig.  93. 

But  0M=  X, 

0X=  arc  PX=  versin"^  QX  to  the  radius  r  =  r  versin^ -, 

r 

and  3IX  =PQ=  VXQQT  =  V2T-^^^'. 

Hence  the  required  equation  is 


-iV 


X  =  r  versin  "~  —^'Zrv 
r 


-V2i 


r 


(1) 


Discussion  of  the  equation.  Since  x  is  imaginary  if  y  is 
negative,  the  curve  lies  wholly  above  X.     If 

y  =  0,  a;  =  r  vers-^0  =  0,  ±  27rr,  ±47rr,  etc., 

or  there  are  an  infinite  number  of  arcs  equal  to  ODA  on  each 
side  of  F,  OA  being  equal  to  the  circumference  of  the  circle. 

If  y—'2  r,  X  —  r  vers~  ^  2  =  ±  7rr,  ±  3  tt/-,  etc. , 

locating  D,  and  the  corresponding  points  on  the  other  arcs. 

Defs.  DD'  is  called  the  axis  of  the  cycloid,  OA  the  base, 
and  0,  A,  etc.,  the  vertices.  To  put  the  generating  circle  in 
position  for  any  point,  as  P,  draw  CC  parallel  to  the  base 
through  the  centre  of  tiie  axis,  and  with  P  as  a  centre  and  a 
radius  =  CD  describe  an  arc  cutting  the  parallel  in  C.  Then  C 
is  the  required  centre.     If  the  angle 

PC'N=  <t>,   0X=  arc  PX=  r<t>. 


TRANSCENDENTAL   CURVES.  187 

a;  =  r(<^-sin<^)l 
and  ^    h  (2) 

which  are  called  the  equations  of  the  c^'cloid,  and  are  more 
useful  than  Eq.  (1)  in  determining  its  properties. 

If  any  other  point  than  P  of  the  radius  of  the  rolling  circle 
be  the  generating  poiut,  the  resulting  curve  is  called  the  jJ^'olate, 
or  curtate  cycloid,  according  as  the  generating  point  is  within 
or  without  the  circle.  The  locus  of  a  point  on  the  circumfer- 
ence of  a  circle  rolling  without  sliding  on  the  circumference  of 
another  is  called  an  epicycloid.,  or  liypocycloid^  according  as  the 
circle  rolls  on  the  exterior  or  interior  of  the  fixed  circle  ;  if  the 
generating  point  is  not  on  the  circumference  of  the  rolling  circle, 
the  curve  is  called  an  epitroclioid  or  hypotrochoid.  The  general 
term  applied  to  the  locns  generated  by  a  point  of  a  rolling  curve 
is  roulette. 

The  circular  functions.  A  series  of  transcendental  curves  is 
obtained  by  assunung  the  ordinate  some  trigonometrical  func- 
tion of  the  abscissa,  as  y  =  sin.'C,  y  =  coto;,  etc.  The  length  of 
the  arc  corresponding  to  any  value  of  x  given  in  degrees  may 
be  found  as  follows  :  The  length  of  the  arc  of  180°  in  the  circle 
whose  radius  is  unity  being  3.1416,  the  length  of  any  other  arc, 
as  that  of  10°,  will  be  J^  (3.1416)  =  .1745  ;  this  distance  being 
laid  off  on  the  axis  of  X,  the  corresponding  value  of  y  may  be 
taken  from  the  table  of  natural  sines,  tangents,  etc.  The  curve 
may  be  drawn,  however,  with  sufficient  accuracy  by  observing 
the  general  change  in  the  function  as  the  arc  increases. 

3.  y=smx.  When  .t  =  0°,  ?/=0,  hence  the  curve  passes 
through  the  origin.  As  x  increases,  y  increases,  reaching  its 
greatest  value  y  =  1  when  x  =  90°,  locating  A.  From  x  =  90° 
to  x=  180°,  y  decreases  from  1  to  0,  becoming  negative  when 
a:>  180°,  and  reaching  its  least  value  y  =  —l  when  x  =  270°, 
locating  C.  From  a:  =270°  to  x'  =  360°,  y  is  negative  and 
decreasing  numerically,  becoming  zero  again  for  x  =  360°.  It 
is  evident  that  as  a;  varies  from  360°  to  720°,  a  like  portion  will 


188 


ANALYTIC   GEOMETRY. 


be  traced,  as  also  when  x  is  negative  ;  hence  the  curve  consists 
of  an  infinite  number  of  arcs  equal  to    OABCD,  and  extends 


Fifi;.  94. 


without  limit  along  X  to  ±  go. 
the  sinusoid. 


The  curve  is  sometimes  called 


4.  y=cosx.     Let  the  student  trace  the  curve. 

5.  y=t2inx.     "When    x=0°,    y  =  0.       As   x   increases,    y 
increases,  becoming  oo  when  x  =  90°.     When  x  passes  90°,  y  is 

negative,  and  remains  negative 
tilhf=180°,  decreasing  numer- 
ically from  00  to  0.  From 
X  =  180°  to  aj  =  270°,  y  is  posi- 
tive and  increasing,  becoming 
oo  when  x-=270°,  etc.  The 
curve  consists  of  an  infinite 
number  of  branches  equal  to 
AOB,  on  either  side  of  the 
origin,  having  for  as3'mptotcs 

q 

the  lines  x  =  ±  —>    x  =  ±  -tt, 
2  2 

Fig.  95.  etc. 

C.    ?/=cot.r.      Let  the   student  trace  tlie  curve. 

7.    ?/  =  versinic.     The  versine  being  always  positive,  the  curve 


lies  wholly  above  X,  its  limits  along  F  being  0  and  2. 
student  trace  the  curve  ;  also  : 

8.    y  =  coversin  x. 


Let  the 


TRANSCENDENTAL   CURVES. 


189 


9.  y=  seca;.     The  curve  is  given  in  the  figure.     Let  the  stu- 
dent discuss  the  equation,  and  also  trace  the  curve : 

10.  y  =  cosecx. 


O 


Fig.  96. 

Spirals.  The  locus  of  a  point  receding  from  a  fixed  point 
along  a  straight  line,  u-hich  revolves  about  the  fixed  point  in  the 
same  plane,  is  called  a  plane  spiral. 

The  fixed  point  is  called  the  pole,  and  that  portion  of  the 
spiral  traced  during  one  revolution  of  the  line  is  called  a  spire. 
The  polar  equations  of  many  of  the  spirals  may  be  derived  from 
the  general  form  r  =  aO'\  by  assigning  different  values  to  n. 

11.  Spiral  of  Archimedes.  The  equation  of  this  spiral  is 
obtained  from  the  general  form  by  making  n—1;  whence 

r  =  a6.  (1) 

From  this  equation-  =  a  ;  since  the  ratio  of  the  radius  vector  to 

the  vectorial  angle  is  constant,  the  spiral  may  be  defined  as 
traced  by  a  point  ivhich  recedes  uniformly  from,  while  the  line 
revolves  uniformly  about,  the  pole.  Assuming  as  a  unit  radius 
the  value  of  r  when  the  line  has  made  one  revolution,  we  have 


l=a.2 


TT 


a  =  — 1 
27r 


and  Eq.  (1)  becomes 


;7r 


(2) 


190 


ANALYTIC   GEOMETRY. 


when  ^  =  0,  r  =  0,  or  the  spiral  begins  at  the  pole.  The  dis- 
tance between  any  two  consecutive  spires  measured  on  the  same 
radius  is  the  same  and  equal  to  the  unit  radius,  called  the  radius 


of  the  measuring  circle,     r  increases  uniformly  with  ^,  but  is  oo 
only  where  ^  =  x . 

To  construct  this  spiral,  let  0  be  the  pole  and  OA  the  polar 
axis.     Through   the  pole  draw  any  number  of  straight  lines 

making  equal  angles  with  each  other,  sav  30°,  or  -•    Then  when 

6 


TT 


6=-, 
6 


V  —     1_ 


r  =  ^'^  =  OP.      Having  laid  off  0P=^  on  01,  make 

OP  =  2  OP,  OP"^i}OP,    etc.       Tlien   OPP'P",  etc.,  is  the 
spiral.     OQ  is  the  radius  of  the  measuring  circle. 

12.  The  reciprocal  ok  hyperbolic  spiral.  The  equation 
of  this  spiral  is  obtained  from  the  general  form  by  making 
Ai  =  —  1  ;  whence 


TRANSCENDENTAL   CURVES. 


191 


In  this  spiral  the  radius  vector  evidently  varies  inverse!}-  as 

the  angle.     Assuming  as  before  that  r  is  unity  when  ^=27r, 

we  have  a  =  27r,  or 

27r  ^2) 


r  = 


6 


When  ^  =  0,  r=  X,  and  as  6  increases,  r  diminishes,  but  is 

r 


Fig.  98. 

zero  only  when  ^  =  oo  ;  hence  there  are  an  infinite  number  of 
spires  between  the  measuring  circle  and  the  pole. 

To  construct  the  spiral,  draw  the  lines  making  equal  angles 
with   each  other,  as  before.     If  the   angle  is  30°,  then  when 

6  =  -,    r=  12=  OP.     Make   OP' =  —  =  6,    OP"  =  ^  =  4, 
6  2  3 

etc.,  and  draw  PP'P"  ••• . 

13.    The  lituus.     This  spiral  corresponds  to  n  =  — i  in  the 
general  equation.     Hence  its  equation  is 


r  = 


a 


-Vd 


(1) 


Fig.  99. 


192 


ANALYTIC   GEOMETRY. 


or,  if  r=  1  when  6=2Tr. 


a 


V27r, 


?-  = 


0 


(2) 


For  every  value  of  6,  r  has  two  values,  one  positive  and  one 
negative,  as  shown  in  the  figure.  If  ^=0,  r  =  cc,  and  r—0 
onl}'  when  $  =;.-'*  ;  there  are  thus  an  infinite  number  of  spires 
between  the  measuring  circle  and  the  pole.  It  may  be  shown 
that  the  polar  axis  is  an  asymptote  to  the  spiral. 

14.    The  logarithmic  spiral.     This  spiral  is  defined  by  the 
equation 

log  r=a^,  (1) 


or,  if  b  be  the  base  of  the  system, 

r  =  &«*. 


(2) 


Whatever  the  logarithmic  system  r  =  l  for  ^=0.      Hence,   if 

0A=  1,  all  logarithmic  spirals 
pass  through  A  ;  and  OA  may 
be  taken  as  the  radius  of  the 
measuring  circle.  From  (2) 
we  see  that  as  6  increases,  r 
increases  rapidly,  and  the  more 
so  as  the  base  is  greater  ;  and 
diminishes  rapidly  if  6  is  nega- 
tive, but  is  zero  only  when 
9  =  —cc.     Also,  if  ^=00, 

r  =  cc. 

-     Hence   there   are    an    infinite 
number  of    spires  within   and 


Fig.  luo. 


without  the  measuring  circle. 


PART   II. 
SOLID   Al^ALYTIO    GEOMETEY. 


CHAPTER   V. 


THE  POINT,  STRAIGHT  LINE,  AND  PLANE. 


-oo^O^oo- 


SECTION   XIV.  —  INTRODUCTORY  THEOREMS. 


132.  Defs.  1.  By  the  angle  betiveen  ttvo  straight  lines  not  in 
the  same  ^jlane  is  meant  the  angle  between  any  tivo  intersecting 
parallels.  Hence,  if  through  any  point  of  one  of  the  lines,  a 
Ijarallel  is  drawn  to  the  other,  the  angle  between  the  first  and 
the  parallel  is  the  angle  between  the  two  lines.  Thus,  let  PQ 
and  KL  be  any  two  straight  lines  which  neither  intersect  nor 
are  parallel.  Through  any  point  of  FQ,  as  P,  draw  FH  par- 
allel to  KL.     Then  HFQ  is  the  angle  between  FQ  and  KL. 

2.  The  foot  of  a  ^perpendicular  from  a  point  upon  a  plane  is 
called  the  projection  of  the  point  on  the  plane.  Thus,  if  F  be 
any  point,  AB  any  plane, 

and  the  perpendicular  to 
the  plane  through  F  meets 
the  plane  at  il/,  M  is  the 
projection  of  F  on  AB. 

3 .  The  foot  of  a  perpen- 
dicxdar  from  a  pioint  on 
a  line  is  the  projection  of 
the  point  on  the  line.  Thus, 
if  KL  be  any  straight  line 
and  the  perpendicular  from 
P  meets  the  line  at  S,  S  is 
the  projection  of  F  on  KL. 


Fig.  101. 


196  ANALYTIC   GEOMETRY. 

4.  The  projection  of  a  straight  line  of  limited  length  upon  a 
plane  is  the  line  joining  the  feet  of  the  j^erj^endiculars  from  the 
extremities  of  the  line  upon  the  plane.  Thus,  if  PQ  be  any 
limited  straight  line,  AB  any  plane,  PM,  QN,  perpendiculars 
to  the  plane  meeting  it  at  M  and  N^  MN  is  the  projection  of 
the  line  PQ  upon  the  plane  AB.  Since  the  perpendiculars  PM 
and  Q^  determine  a  plane  through  PQ  perpendicular  to  AB, 
the  projection  of  a  straight  line  upon  a  plane  ma}^  also  be 
defined  as  the  intersection  of  a  plane  through  the  line,  perpen- 
dicular to  the  given  plane,  with  the  latter. 

5.  Tlie  p>rojection  of  a  limited  straight  line  upon  another  straight 
line  is  that  portion  of  the  latter  intercepted  bj/  the  j)i'ojections  of 
the  extremities  of  the  former.  Thus,  PS  and  QT  being  the 
perpendiculars  from  P  and  Q  on  KL,  JST  is  the  projection  of 
PQ  on  KL. 

133.  The  projection  of  a  limited  straight  line  upon  another 
straight  line  is  equal  to  the  length  of  the  line  multiplied  by  the 
cosine  of  the  included  angle. 

The  projections  of  any  limited  straight  line  upon  parallels  are 
equal ;  for  the  perpendiculars  from  its  extremities,  being  per- 
pendicular to  parallel  lines,  lie  in  parallel  planes  ;  these  planes, 
therefore,  intercept  equal  distances  on  the  parallels,  and  these 
distances  are  the  projections.  Thus,  in  Fig.  101,  KL  and  3fN 
being  parallel,  the  planes  PS M aud  QTN  are  parallel,  and  the 
intercepts  *S'Tand  3/iVare  equal.  Hence,  if  we  find  the  pro- 
jection of  PQ  on  any  one  of  a  system  of  parallels,  this  projec- 
tion will  be  the  same  for  all.  Draw  P^  parallel  to  MN.  The 
angle  between  PQ  and  any  parallel  to  PH  is  (Art.  132,  1)  HPQ, 
and  IIP  =  PQ  cos  HPQ  =  MN=  ST=  etc.  Hence  the  propo- 
sition. 

CoR.  Since  the  angle  between  PQ  and  AB  =  NRQ  =  HPQ, 
the  projection  of  a  limited  straight  line  upon  a  p>lane  is  equal  to 
the  length  of  the  line  multiplied  by  the  cosine  of  the  angle  which 
the  line  makes  with  the  plane.     Thus  MN=  PQ  cos  NRQ. 


INTRODUCTORY  THEOREMS. 


197 


134.  If  AD  he  the  straight  line  joining  A  tvith  i>,  and  AB, 
BC,  CD,  straight  lines  forming  a  broken  line  from  A  to  D,  then 
the  algebraic  sum  of  the  projections  of  the  latter  upon  any  straight 
line  OX  is  equal  to  the  projection  of  the  former  on  the  same  line. 
Draw  from  A,  B,  C,  aud 
D,  the  perpendiculars  to 
OX,  meeting  OX  in  A', 
B',  C,  D',  respectively. 
It  is  evident  that  as  A 
moves  to  D  along  the 
bi'okeu  line  ABCD,  the 
foot  A',  of  the  perpendic- 
ular from  A,  moves  along 
A'D\  to  the  right  or  the  left,  according  as  the  angle  between  the 
direction  of  motion  of  A  and  OX  is  acute  or  obtuse.     At  -4, 

B,  and  C,  draw  parallels  to  OX,  and  denote  the  angles  made 
by  AB,  BC,  CD,  and  AD  with  these  parallels  by  a,  (3,  y,  and  8. 
As  the  points  are  chosen  in  the  figure,  in  passing  from  B  to 

C,  B'  moves  to  the  left  along  OX,  the  angle  B  being  obtuse. 
Now  the  projection  on  OX  of  AB  is  (Art.  133) 

A'B'^AB  cos  a; 
that  of  BC  is  B'C  =  BC  cos  (3 ',~ 

that  of  CD  is  CD'  =  CD  cosy  ; 

and  the  algebraic  sum  of  these  projections  is 

A'B'  -  B'C  +  CD', 

since  cos^S  is  negative.  But  this  sum  is  A'D',  which  is  also 
equal  to  AD  cos  8,  or  the  projection  of  AD  on  OX. 

Hence  AB  cos  a -{-  BC  cos  (3  +  CD  cos  y  =  AD  cos  8. 

The  same  will  evidently  be  true  if  we  take  any  number  of 
lines  between  A  and  D.  Hence,  if  two  given  pioiMs  are  joined 
by  a  broken  line,  the  algebraic  stan  of  the  projections  of  its  parts 
upon  any  given  straight  line  is  equal  to  the  projection  on  the  same 
line  of  the  straight  line  joining  the  two  given  points. 


198 


ANALYTIC   GEOMETRY. 


SECTION  XV.  —  THE  POINT. 


135.  Position  of  a  point  in  space.  The  position  of  any  point 
in  space  may  be  determined  by  referring  it  to  three  fixed  planes 
meeting  in  a  point.      Thus,  if   XOY,    YOZ,  ZOX^  be   three 

planes  meeting  at  0.  and 


intersecting  each  other 


ni 


Fig.  103. 


the  lines  OX,  0  F,  OZ,  the 
position  of  P,  relativeh^  to 
tliese  planes,  will  be  known 
when  its  distances  PQ,  PR, 
PS,  from  each,  measured 
parallel  to  the  other  two, 
are  known.  The  three 
l)laues  are  called  the  Coor- 
dinate Planes,  their  three 
lines  of  intersection  the 
Coordinate  Axes,  their  common  point  the  Origin,  and  the  dis- 
tances PQ,  PR,  PS,  the  Coordinates  of  P. 

If  the  planes,  and  consequently  the  axes,  are  at  right  angles 

to  each  other,  the  coordi- 
nates are  said  to  be  rectan- 
gular ;  otherwise  they  are 
oblique.  Use  will  be  made 
only  of  rectanoular  coor- 
dinates,  and  they  will  there- 
fore be  in  all  cases  the 
perpendicular  rh'stances  of 
the  point  from  the  coor- 
dinate planes. 

It  is  customary  to  assume  the  axes  as  in  the  figure,  the  plane 
XOF  being  horizontal  and  the  axis  OZ  vertical.     For  brevity, 


Fig.  104. 


THE   POINT.  199 

the  coordinate  planes  will  be  referred  to  as  the  planes  XY,  YZ, 
and  ZX,  and  the  coordinate  axes  as  the  axes  of  X,  Y,  and  Z. 
The  coordinates  PQ,  PR,  P*S,  are  represented  by  the  letters 
X,  y,  z,  corresponding  to  the  axes  to  which  they  are  parallel. 

The  coordinate  i)lanes  divide  space  into  eight  right  triedral 
angles  which  are  numbered  as  follows  :  the  Jirst  lies  above  XY, 
to  the  right  of  YZ,  and  in  front  of  ZX;  the  second  to  the  left 
of  the  first ;  the  third  behind  the  second  ;  the  fourth  behind  the 
first;  the Jlfth,  sixth,  seventh,  and  eighth,  lying  under  the  first, 
second,  third,  and  fourth,  in  order. 

If  we  extend  to  Z  the  convention  of  signs  already  adopted 
for  X  and  F,  the  positive  direction  of  Z  being  upward,  it  is 
evident  that  while  the  absolute  values  of  x,  y,  2,  may  be  the 
same  for  dilferent  points,  their  signs  will  determine  in  which 
of  the  eight  angles  any  given  point  lies,  and  that  a  point  will 
thus  be  completely  determined  when  its  coordinates  are  given 
in  magnitude  and  sign. 

136.  Equation  of  a  point.  The  position  of  a  point  may  be 
designated  by  the  equations  a^  =  a,  y=b,  z=  c  ;  or  b}'  the  nota- 
tion (a,  b,  c),  the  coordinates  being  written  in  the  order  x,  ?/,  z. 
To  construct  the  point  {x,  y,  z),  construct  the  point  (x,  y),  S 
of  Fig.  104,  in  the  plane  XY,  and  at  S  erect  the  perpendicular 
SP=z. 

Examples.  1.  In  what  angle  is  the  point  («,  —b,  — c)  ? 

2.  Write  the  coordinates  of  a  point  in  each  of  the  eight  angles. 

3.  In  what  plane  is  the  point  (o,  b,  0)  ? 

4.  "Write  the  coordinates  of  a  point  in  each  of  the  three  coor- 
dinate planes. 

5.  To  what  plane  is  the  point  (x,  y,  c)  restricted? 

Ans.   To  a  plane  parcdlel  to  XY,  at  a  distance  cfrom  it. 

6.  What  are  the  coordinates  of  points  in  a  plane  parallel  to 
YZ  at  a  distance  a  from  it? 


200 


ANALYTIC   GEOMETRY. 


7.  Where  is  the  point  (x,  —b,  z)? 

8.  On  what  axis  is  the  point  (a,  0,  0)  ? 

9.  Write   the  coordinates  of  a  point   on   each  of   the  three 
coordinate  axes. 

10.  What  are  the  coordinates  of  the  origin? 


137.   Distance  between  two  given  points. 
Let  x\  y',  z',  be  the  coordinates  of  P' ;  x",  y",  z",  those  of  P". 

Then 

But 

Hence 


P'P"  =  VP'IP  +  KP"^. 
P'lP  =  P'H-  +  HK'. 

P'P"  =  ^'P'H'  +  HIP  +  KP'"' 


Now 

Similarly, 
Hence,  if 


P'H =2^^'  =  "^'-  ~~  ^P  =^"  —  ^'• 
HK=  y"  -  y',  KP"  =  z"  -  z'. 
P'P"  =  D, 


'\2 


D  =  V{x"-x'y  +  {y"  -  y'y+  {z"  -z'J 

Cor.  1.     If  one  of  the  points,  as  P",  is  at  the  origin, 

x"  =  y"  =  z"  =  0. 
Heuce  the  distance  of  a  point  {x\  y',  z')  from  the  origin  is 

D  =  ^x'-'  +  y'-  +  z'\ 


THE   POINT.  201 

Cor.  2.     If  z'  =  0,  that  is,  if  P'  is  at  p  in  the  plane  XY, 


D  =  Vx'^  +  y'- 

is  the  distance  of  j)  from  the  origin  =  distance  of  P'  from  the 
axis  of  Z.     Hence  the  distances  of  {x',  y',  z')  from  the  axes  of 
X,  Y,  Z,  are         y^TqT^^     V.^?M^,     V^+F, 
respectively. 

138.  Polar  coordinates  of  a  point. 

The  position  of  a  point  in  space,  may  also  be  determined  by 
polar  coordmates.  For  this  purpose  assume  any  fixed  plane,  as 
XY  (Fig.  104),  and  any  fixed  line  in  that  plane,  as  OX,  0 
being  the  pole.  Then  the  position  of  P  will  be  determined  when 
we  know  OP,  its  distance  from  the  pole  ;  the  angle  SOP,  which 
OP  makes  with  the  plane  XY;  and  the  angle  /SOX  which  the 
line  SO  makes  with  X.  OP  is  called  the  radius  vector  of  P 
and  is  represented  by  r,  OS  is  the  projection  (Art.  134,4)  of 
OP  on  XY,  the  angle  ^SOP  being  represented  by  <^,  and  ZOS 
by  6.  The  point  Pmay  then  be  designated  as  the  point  (;-,  6,  (jy) , 
$  and  cji  determining  its  direction,  and  r  its  distance,  from  the 
pole  0. 

139.  Relations  between  polar  and  rectangular  coordinates. 
In  Fig.  104,  we  have, 

X  =  OL  =  OS  cos  6  =  OP  cos  ^  •  cos  0  =  r  cos  ^  cos  6,  ( 1 ) 
y=  LS  =  OSsmO  =  OP  cos  <^.  sin^  =  r  cos  <^  sin  ^,  (2) 
2  =  P>S' =  OP  sine/)  =  'r  sin  (/>;  (3) 

the  rectangular  in  terms  of  the  polar  coordinates. 
From  Art.  137,  Cor.  1,  we  have 


op=r  =  V.T-  +  r  +  2';  (4) 

from  the  triangle  OLS. 

SL     y  .rx 

tan  0  =  Trr^  =  ■-  ;  V.^ J 
OL      X 


202 


ANALYTIC   GEOMETRY. 


from  the  triangle  SOP, 
tan^: 


PS  ^         z 

OS     Vx-N^^ 


(6) 


the  polar  in  terms  of  the  rectangular  coordinates. 


140.   Direction  angles  and  cosines. 

The  angles  made  by  any  straight  line  ivith  the  axes  are  called 
its   direction-angles.     Since   parallel   lines   make  equal  angles 


with  the  axes,  draw  OP,  parallel  to  the  given  line,  through  the 
origin.  Then  LOP  =  a,  3I0P  =  (3,  NOP=  y,  are  the  direction- 
angles  of  OP,  or  of  any  parallel  to  OP,  and  are  always  esti- 
mated from  the  positive  directions  of  the  axes.  The  cosines  of 
the  direction-angles  are  called  the  direction-cosines. 

141.    The  sum  of  the  squares  of  the  direction-cosines  of  any 
straight  line  is  unity. 

Join  P,  Fig.  lOG,  with  L.     Then,   since  the   plane  PSL  is 
perpendicular  to  OX,  the  triangle  PLO  is  right-angled  at  L, 

and  OL  =  x  =  OP  cos  L  OP  =  r  cos  a  ; 

similarly,  drawing  PM  and  PN, 

OM  =y=OP  cos  MOP  =  r  cos  13, 
ON  =  z=OP  cos  NOP  =  r  cosy. 


THE   POINT.  203 

Squaring  aud  adding, 

OL^  +  OM-  +  ON^  =  r  (cos^a  +  cos-/3  -f-  cos^y) . 
But  the  first  member  is  r  (Art.  137,  Cor.  1).     Hence 
COS^a  +  COS-^  +  COS-y  =  1. 

Examples.  1.  Two  of  the  direction-angles  of  a  straight  line 
are  60°  and  45°.     Show  that  the  third  is  60°. 

2.  Find  the  distances  of  (4,  -7,  4)  from  the  axes,  and 
from  the  origin. 

3.  Find  the  distance  of  (4,  -2,  -1)  from  (6,  3,  2). 

142.  To  find  the  angle  betiveen  two  straight  lines  ivhose  direc- 
tion-cosines are  given. 

Let  OP,  OQ,  be  parallels  to  the  given  lines  through  the  origin, 
and  let  a,  /3,  y,  and  a',  (3\  y',  be  the  angles  made  by  OP  and 
OQ,  respectively,  with  the  axes  of  X,  F,  and  Z.    These  angles 


T^h—x 


Fig.  ]07. 


are,  then,  the  direction-angles  of  the  given  lines.  Let  QOP=  0, 
and  x,y,  z,  be  the  coordinates  of  Q.  Then  (Art.  134),  the 
projection  of  OQ  upon  OP  is  equal  to  the  algebraic  sum  of  the 
projections  of  OL,  LS  and  SQ,  upon  OP.  But  the  projection 
of  Oi.  on  OP  is  (Art.  133) 

0Z(  COSa=  OQ  COSa'-COSa. 


204  ANALYTIC    GEOMETRY. 

Similarly,  the  projections  of  LS  and  SQ  on  OP  are 
LS  cos (3=  OQ  cos;8'-cosj8, 

SQ  cosy  =  OQ  cosy'  •  cosy. 
The  projection  of  OQ  on  OP  is  OQ  cos  0.     Hence 
0Qcos6  =  0Q  cosa'  cosa+  OQ  cos/5'  cos/3+OQcosy'  cosy, 

or         C0S^=  cos  a' cos  a  +  cos  y8  cos/?'  +  cosy  cosy', 

or  the  cosine  of  the  angle  inclnded  between  any  two  straight 
lines  is  the  sum  of  the  rectangles  of  their  corresponding 
direction-cosines. 


THE   PLANE. 


205 


SECTION    XVI.— THE   PLANE. 


143.   General  equation  of  a  surface. 

We  have  seen  that  any  point  (x,  y,  z)  may  be  constructed 
by  first  locating  the  point  (.x,  y)  in  the  plane  XF",  and  then 
laying  off,  on  a  perpendicular  through  this  point  to  XI^,  the 
distance  z.  Hence,  if  z  be  made  equal  to  any  constant,  as  a, 
X  and  y  remaining  variables,  the  point  (.r,  y,  a)  will  lie  in  a 
plane  parallel  to  XY.     If,  therefore, 

f{x,y,z)  =  0  (1) 

be  any  equation  between  x-,  y,  and  2,  and  in  this  equation 
2  =  a,  a  constant,  then  /(a;,  ?/,  a)  =  0,  being  an  equation 
between  two  variables,  will  represent  a  line,  all  of  whose 
points  are  in  a  plane  parallel  to  XY  at  a  distance  a  from 
it.  Similarly  if  z  =  b,  f{x,  y,  h)  —  0  will  be  the  equation 
of  a  line  in  a  plane  parallel  to  XY^  at  a  distance  b  from  it. 
Giving  thus,  successively,  to  z,  all  possible  values,  i.e.,  letting 
z  vary  continuously  between  the  limits  assigned  b}'  the  equation 
f{x,  y,  z)  =  0,  we  obtain  a  series  of  lines,  all  of  which  are  plane 
curves  parallel  to  XY",  which, 
taken  together,  form  a  sur- 
face of  which /(a.*,  y,  z)  =  0 
is  the  equation.  Hence 
f{x,  y,  z)  =  0  is  the  equa- 
tion of  a  surface. 

To  illustrate:  let  P  be 
any  point  in  space  subject 
to  the  condition  that  the  sum 
of  the  squares  of  its  coor- 
dinates is  a  constant,  or 

X^+f+z'^R'.       (2)  -  Fig.  108. 


200  ANALYTIC    GEOMETKY. 

Since  this  sum  is  the  square  of  the  distance  of  P  from  the 
origin  (Art.  137),  it  is  evident  that  P  is  restricted  to  the  sur- 
face of  a  sphere  whose  radius  is  R  and  wliose  centre  is  at 
the  origin,  and  of  which  (2)  is  the  equation.  If  we  assume 
2  =  0,  then  ar+i/-  =  J?-  is  the  equation  of  a  circle  in  the  plane 
Xy,  whose  radius  is  that  of  the  sphere  ;  that  is,  it  is  the  great 
circle  cut  from  the  sphere  by  XY.  If  we  make  z  =  a,  we  have 
oi?-\-'if=  Pr—a^,  which  is  also  the  equation  of  a  circle,  namely, 
that  cut  from  the  sphere  by  a  plane  parallel  to  XY  at  a  dis- 
tance from  it  equal  to  a.  As  a  increases,  the  radius  of  the 
circle,  V-B^—  or-,  diminishes,  and  when  z  is  made  equal  to 
a  =  P,  we  have 

a^+  y^  =  0,    or   x=  0,    ?/  =  0, 

the  plane  then  touching  the  sphere  at  its  highest  point  (0,  0,  P). 
z  cannot  be  made  greater  than  P,  for  then  x- -\- y~  —  P- ~  a^ 
would  be  impossible,  since  the  sum  of  two  squares  cannot  be 
negative,  showing  that  no  plane  at  a  greater  distance  from  XY 
than  P  can  cut  the  surface  of  the  sphere. 

The  lines  cut  from  an\'  surface  by  a  plane  are  called  sections 
of  the  surface.     If  x  were  made  constant  in  (2),  then 

would  be  the  section  cut  by  a  plane  parallel  to  YZ  from  the 
surface  of  the  sphere  ;  and  if  y  were  made  constant,  we  should 
have  the  section  made  by  a  plane  parallel  to  ZX,  all  of  which 
would  in  this  case  be  circles.  And,  in  general,  if  in  the  equa- 
tion of  any  surface,  f{^i  Vi  ^)  =  0,  one  of  the  variables  he  made 
constant,  the  resulting  equation  is  that  of  the  line  cut  from  the 
surface  by  a  plane  j)araUel  to  the  jilane  of  the  other  two  axes  and 
at  a  distance  from  it  equal  to  the  value  assigned. 

144.  Equation  of  a  plane.  If,  when  either  .r,  y,  or  z  is 
made  constant  in  the  equation  f(x,y,z)=0,  the  resulting 
equation  is  of  the  first  degree  between  the  two  remaining 
variables,  every  section  of  the  surface  by  planes  parallel  to  the 
coordinate  planes  is  a  straight  line,  and  the  surface  must  be  a 


THE   PLANE. 


207 


plane.  But  this  can  be  the  case  only  when/(ic,  ?/,  z)  =  0  is  of 
the  first  degree  with  respect  to  all  three  of  the  variables. 
Hence,  every  equation  of  the  form 

Ax  +  By+Cz  +  F=0  (1) 

is  the  equation  of  a  plane. 


145.    Intercept  form  of  the  equation  of  a  plane. 

If  in  the  equation  of  a  plane, 

Ax  +  By+Cz  +  F=Q, 

F 


(1) 


we  make  y  =  z  =  0,  we  obtain  x  =  OQ  =  —  ^,  the  intercept  of 

F 

the  plane  on  X.     Making  z  =  x  —  0,  we  have  y=zOR= 


F 


B' 


the  intercept  on  Y;    and  for  .i-  =  ?/  =  0.  z  =  OS  =  —  ^,  the  in- 

tercept  on  Z.     Representing  these  intercepts  by  a,   b,  and  c, 
respectively,  we  have 


F     ,           F 

F 

c 

whence 

A  =  -^,B  =  -^,C  = 

a                b 

F 

—  —  • 

c 

Substituting    these    values 

in 

(1),  we  have 

B 

a      b      c 

(2) 

-^ 

Fig.  109. 


the  equation  of  a  plane  in  terms  of  its  intercepts. 

This  form  is  not  applicable  when  the  plane  passes  through 
the  origin  ;  for  in  this  case,  since  the  origin  is  a  point  of  the 
plane,  (0,  0,  0)  must  satisfy  its  equation,  and  from  (1), 
F=0,  and  a  =  b  =  c  =  Q. 

To  put  the  equation  of  a  plane  in  the  intercept  form,  trans- 
pose the  absolute  term  to  the  second  member,  and,  by  division, 
make  the  second  member  positive  unity.     Thus,  the  intercept 


208  ANALYTIC    GEOMETRY. 

0'  z 

form  of  3x  —  Gy  +  2z  —  6  =  0  is  '— ?/+=!,  the  intercepts 
being  2,  —1,  3. 

To  write  the  equation  of  a  plane  whose  intercepts  are  given, 
substitute  their  values  in  (2).  Thus,  the  equation  of  the  plane 
whose  intercepts  are  4,  —3,  —6,  is 

--^--=1,     or     3.r-4w-2z-12  =  0. 
4      3      6'  ^ 

Examples.  1.  Write  the  equation  of  a  plane  whose  inter- 
cepts are  2,  G,  4;   also  —2,  —3.  1. 

Alls.   6x  +  2y  +  Sz-12=0;    Sx +  2y  -  Qz-\- 6  =  0. 

2.  Put  the  following  equations  under  the  intercept  form  : 

OX  +  oy  —  z  +  15  =  0,    X  —  y  —  z—  1  =  0. 

3.  Determine  the  intercepts  of  the  planes  of  Ex.  2,  without 
putting  the  equations  under  the  intercept  form. 

146.  Normal  equation  of  the  plane.  Let  QRS,  Fig.  109, 
be  any  plane  ;  OD  a  perpendicular  upon  it  from  the  origin,  its 
length  being  p,  and  a,  ^,  y,  its  direction-cosines.  Let  P  be  any 
point  of  the  plane,  .t,  y,  z,  being  its  coordinates.  Then  the  pro- 
jection on  OD  of  OP  is  equal  to  the  sum  of  the  projections  of 
ON,  NM,  and  3fP,  on  OD  (Art.  134).  But,  whatever  the 
position  of  P  in  the  plane,  the  projection  of  OP  on  OD  is  j3, 
since  OD  is  perpendicular  to  the  plane  ;  and  the  projections  of 
ON,  NM,  MP,  are  a;  cos  a,  ycosf3,  2  cosy  (Art.  133). 

Hence  ajcosa-j- ?/cos/3 +  2:  cosy  =p  (1) 

is  the  normal  equation  of  the  plane,  in  which  }'>  is  always  posi- 
tive. Since  the  sum  of  the  squares  of  the  direction-cosines  of 
any  line  is  unity,  to  put  the  equation  of  a  plane  under  the  nor- 
mal form,  we  must  introduce  a  factor  B  fulfilling  the  condition 

(BAY  +  (BB)  '  +  {BC)-=l, 

in  which  A,  B,  C,  are  the  coefficients  of  x,  y,  and  z,  in  the 
given  equation  ;  hence 


-J  A'  +  i3=  +  C 


THE   PLANE.  209 


Thus,  the  normal  form  of  ox  +  2y  —  z+\  =0  is 
Sx         2y  2    _     1 


Vl4      Vl4      Vl4      Vl4 
the  second  member  being  made  positive  ;  in  which 

3  2  1 


-? 


Vl4  Vl4      Vl4 

are  the  direction-cosines,  and  p  =  — —  =  distance  of  the  plane 
from  the  origin.  V 14 

Examples.  1.  Put  the  equations  ox-\- 5y  —  z-\-15  =  0, 
x  —  y  —  z—\—  0,  under  the  normal  form. 

2.  Find  the  distance  of  the  plane  x  —  y-\-z  —  l  =  0  from  the 
origin.     In  what  angle  is  the  perpendicular  from  the  origin  on 

the  plane .  Aus.  — -  ;  in  the  fourth  angle. 

V3 

3.  Show  that  4.t  +  7?/ +  4^  —  9  =  0  is  at  a  distance  nnity 
from  the  origin. 

4.  Write  the  equation  of  a  plane  whose  distance  from  the 

origin  is  10,  the  direction-cosines  of  the  perpendicular  being 

1     1      V23  ,- 

-'    -'    — r—  Ans.  33;  +  2?/  + V232-G0  =  0. 

5.  Are  the  direction-cosines  of  Ex.  4  chosen  at  random? 

6.  Write  the  equation  of  a  plane  parallel  to  YZ  at  a  distance 
from  YZ  equal  to  6. 

Since  the  plane  is  parallel  to  YZ,  its  intercepts  on  Y  and  Z  are  both 
infinity.  Hence,  from  Eq.  2,  Art.  113,  b—co,  c  =  aD,  and  x  :=  a  =  6. 
Or,  from  Eq.  1,  Art.  114,  a  =  0,  ^  =  -y  =  90° ;  hence  x  =  p  =  6. 

7.  Write  the  equation  of  a  plane  parallel  to  X. 

8.  What  are  the  equations  of  the  coordinate  planes? 

147.  To  write  the  equation  of  a  plane  through  three  given 
points.     Assuming  the  general  equation 

Ax-\-By-\-Cz  +  F=:0,  (1) 


210  ANALYTIC   GEOMETllY. 

dividiDg  by  any  one  of  the  four  constants,  as  F^  and  denoting 
the  resulting  coefficients  by  A',  B',  C,  we  obtain 

A'x  +  B'y+C'z+\=0.  (2) 

Substituting  in  this  equation  the  coordinates  of  the  three  given 
points  in  succession,  there  results  three  equations  between  A', 
B',  and  C",  from  which  the  values  of  these  latter  may  be  deter- 
mined. Substituting  these  values  in  (2),  we  have  the  equation 
required. 

Examples.  Write  the  equations  of  the  planes  through  the 
following  points. 

(1,0,-2),  (3,2,-1),  (5,-1,2).  .4ns.  9x-4y-10z-20  =  0. 

(1,2,3),      (4,5,6),      (-7,8,9).  Ans.  y-z-{-l  =  0. 

(0,0,0),     (1,2,4),     {1,-2,6).  Ans.   Wx-y-2z  =  0. 

(2,0,0),     (0,2,0),      (0,0,2).  Ans.  x  +  y  +  z-2=0. 

(0,1,2),      (0,2,4),      (1,0,2).  A71S.  2x  +  2y-z=0. 

148.  To  find  the  angle  betiveen  tico  given  planes. 

The  angle  between  the  planes  is  the  same  as  that  between  the 
perpendiculars  upon  the  planes  from  the  origin.  Hence,  if 
a,  yS,  y,  and  a',  /3',  y',  are  the  direction-angles  of  the  perpendicu- 
lars, and  6  the  angle  between  the  latter,  the  required  angle  is 
given  by  the  relation  (Art.  142) 

cos  6  =  cos  a  cos  a'  +  COS  /?  COS  P'  -f-  cos  y  cos  y'.  ( 1 ) 

If  the  equations  of  the  planes  are  given  in  the  normal  form, 
we  have  only  to  substitute  in  (1)  the  coefficients  of  the  variables, 
they  being  the  direction-cosines  (Art.  146).  If  the  equations 
are  in  the  general  form 

Ax  -f  By  +  C^  +  F=  0,  A'x  +  B'y  +  C"z  +  F'  =  0, 
their  normal  forms  will  be 

Ax  +  By  +  Cz     ^  F 

VA^B^+C'        y/A^+B'-{-C'' 
A'x  +  B'v  +  C'z  F' 


VA"+ B'-  -t-  C '-      VA''  +  B'''  +  C' 


THE   PLANE.  211 

in  which  the  radicals  have  the  opposite  signs  of  the  absolute 
terms  in  order  to  make  the  second  members  positive  ;  and  the 
direction-cosines  are,  respectively, 

ABC 


—  ? 


■VA'+B'-\-C'       -JA'  +  B'+G^      ^A'  +  B'  +  C 
A'  B'  C 


Substituting  these  values  in  ( 1 ) , 


(2) 


.  AA'  +  BB'  +  CC  ,., 

cos6'= —  — •  (3) 

VyP  +  B'-\-C'     VA'-  +  B"+C" 

Cor.  1.  If  the  given  planes  are  perpendicular  to  each  other, 
$  =  90°,  cos^  =  0  ;  hence  the  condition  of  perpendicularity  is 

AA'  +  BB'  -\-CC'  =  0,  (4) 

or  the  S}ims  of  the  rectangles  of  the  coefficients  of  the  correspond- 
ing coordinates  in  the  equations  of  the  planes  ninst  he  zero. 

Cor.  2.  If  the  planes  are  parallel,  the  direction-cosines  of 
their  perpendiculars  with  respect  to  each  axis  must  be  equal. 
Hence,  from  (2) 

A'     B'      C'  ^  ^ 


smce  each  ratio  is  equal  to- 


^A'-+B''+C" 

Hence  the  condition  of  parallelism  is  that  the  ratios  of  the  coef- 
ficients of  the  corresponding  coordinates  in  the  equations  of  the 
planes  must  be  equal. 

Examples.     1.  Find  the  angle  between  the  planes  x  -\-2y  — 
22  +  1  =  0  and  307 -j-Gt/- 62-5  =  0.  Ans.  0°. 

2.  Find  the  angle  between  the  planes   2x-\-2y +  z +  1  =  0, 
Ax  —  Ay  +  7z-l  =  0.  Ans.  6  =  cos~^^. 

3.  Show  that  3/^a;  +  ?/- ^^  +  1  =  0  and  12  (.r-j-y) -3^+10  =  0 
are  parallel. 


212  ANALYTIC   GEOMETRY. 

4.  Show  that  xj-2y  —  2z-{-l  =  Ois  perpendicular  to2x  +  5y 
-f  62  —  11=0  ;  also  x-\-2y-\-3z  +  l=0  to  3x-\-6y— oz —o  =  0. 

5.  Write  the  equation  of  a  plane  parallel  to  3.^  +  4?/  — 2 -f- 
6  =  0. 

6.  Write  the  equation  of  a  plane  perpendicular  to  3.T  +  4y+ 
z-l  =  0. 

7.  Find  the  distance  between  the  parallel  planes  x  +  2y  — 
2z  +  l  =  0,  5x  +  6y-6z- 25  =  0.  Ans.  2^. 

8.  Prove  that  Ax-hBy  +  Cz-hF-j-k{A'x+B'y+C'z -{-F')  =  0 
is  the  equation  of  a  plane  through  the  intersection  of  the  planes 

Ax  +  By-\-Cz  +  F=0    and    A'x  +  B'y  +  C'z  +  F'=0. 

See  Art.  37. 

9.  Write  the  equation  of  any  plane  through  the  intersection 
ot  2x  +  5y-\-z—l=0   and   x  —  y  +  z  +  2  =  0. 

10.  Explain  how  to  determine  A;  in  Ex.  8  so  that  the  plane 
shall  pass  through  a  given  point.     See  Art.  37. 

11.  Write  the  equation  of  a  plane  through  the  intersection  of 
2x  +  y  —  z-\-l  =  0  and  oa;4-4?/  +  22;  +  G  =  0,  passing  also 
through  the  point  (1,  1,  2).  Ans.  28iB  + 9^  — 212  +  5  =  0. 

12.  Write  the  equation  of  a  plane  through  the  intersection  of 
the  planes  of  Ex,  11,  and  also  passing  through  the  origin. 

Ans.  9x  +  2y-8z  =  0. 

13.  Prove  that  the  distance  from  the  point  (x',  ?/',  z')  to  the 
plane  a;  cos  a  +  ycosf3  +  2:cosy  =p,  is 

x'  cos  a  +  y'  cos;8  +  z'  cos  y  —  p, 

Ax'-{-Bx'-{-  Cz'  ~F  c       A  ^   QQ 

or  '     —  See  Art.  38. 

VA'  -hB'+C 

14.  Find  the  distances  from  the  plane  5x-i-2y  —  lz-\-d  =  0 
of  the  points  (1,  -  1,  3)  and  (3,  3,  3). 

15.  Find  tlie  equation  of  a  plane  through  (1,  10,  —2),  par- 
allel to  the  plane  2x-{-y  —  z-\-6  =  0.      Ans.2x-{-y  —  z  —  l4:=0. 


THE   PLANE.  213 

16.  Fiud  the  equation  of  a  plane  through   (1,-1,3)  per- 
pendicular to  the  plane  2a;  +  ?/  —  z  +  6  =  0. 

17.  Find   the   distance    from   (8,  14,8)   to  4x+7y  +  4z  — 
18  =  0.  Ans.  16. 

149.    Traces  of  a  plane.      If   in  the  equation  of  a   plane. 
Ax  +  By  -{-Cz  -{-  F=  0^  we  make  z  =  0,  the  resulting  equation 

Ax  +  By  +  F==0  (1) 

applies  to  all  points  of  the  plane  in  XY,  and  is  therefore  the 
equation  of  EQ  (Fig.  109),  the  intersection  of  the  plane  with 
XY.     For  like  reasons 

B>/  +  Cz  +  F=0, 

Ax-^Cz  +  F=0, 

are  the  equations  of  the  intersections  ES  and  SQ.  These  inter- 
sections are  called  the  traces  of  the  plane.     Solving  (1)  for  y, 

we  have 

A        F 

y  = x ,      \ 

B        B 
and  the  corresponding  trace  for  any  other  plane  would  be 

.1'        F' 

y  — X . 

•^  B'        B' 

If  the  traces  are  parallel,  —  =  — .  or  —  =  —     If    the    corre- 
^  B      B'         A'      B' 

sponding  traces  on  the  other  coordinate  planes  are  also  parallel, 

in  which  case  the  planes  themselves  are  parallel,  we  obtain  in 

like  manner 

A'      B'      C' 
the  condition  already  found. 


214  ANALYTIC   GEOMETRY, 


SECTION   XVII.— THE    STRAIGHT  LINE, 


150.   Equations  of  the  straight  line. 

Assuming  the  equation  of  a  plane  in  the  intercept  form, 

-  +  7  +  -=l, 
a      0     c 

if  we  impose  the  condition  that  the  plane  shall  be  perpendicular 
to  XZ,  its  ^-intercept  b  will  be  infinity,  and  its  equation 
assumes  the  form 

Ax  +  Cz-i-F=0,  (1) 

whatever  the  value  of  y.  Hence  every  equation  of  the  first 
degree  between  two  variables  represents  a  plane  perpendicular 
to  the  corresponding  coordinate  plane,  the  third  variable  being 
indeterminate.     Therefore 

X  being  indeterminate,  represents  a  plane  perpendicular  to  YZ. 
Let  ABDL  be  the  plane  represented  by  (1),  and  AHDC  that 
represented  by  (2).  V^alues  of  x,  i/,  z,  which  satisfy  both  (1) 
and  (2)  locate  a  point  in  both  planes,  that  is,  on  AD,  their 
line  of  intersection.  Hence,  while  taken  separately  (1)  and 
(2)  are  equations  of  planes  perpendicular  respectively  to  XZ 
and  YZ,  if  taken  together  they  represent  a  straight  line  in  space. 
Thus,  let.T])e  the  independent  variable,  and  an}'  value  asx  =  03f 
be  substituted  in  (1).  From  (1)  we  may  then  find  z  =  MS  and 
locate  the  point  S  in  XZ.  Now  so  long  as  (1)  is  considered 
independent  of  (2),  it  represents  the  plane  LABD.  and  the 
value  of  y  may  be  assumed  at  pleasure.  But  if  (1)  and  (2) 
are  simultaneous,  y  must  be  derived  from  (2)  after  the  value 
z  =  MS  has  been  substituted  in  it.     Since  this  value  of  y  satis- 


THE   STRAIGHT    LINE. 


215 


fies  (2) ,  y  =  SP  must  locate  a  point  P  iu  the  plane  AIIDC,  and 
P  must  lie  on  the  intersection  AD. 


jj 


Fi2.  110. 


The  Vine  AD  is  evidently  completely  determined  by  (1)  and 
(2),  since  the  planes  can  intersect  in  but  one  straight  line. 

Since  (1)  is  true  for  all  values  of  y,  it  is  true  for  y=  0,  and  is 
then  the  equation  of  the  trace  AB  o^ABDL  on  XZ.  Hence  (1) 
is  the  equation  of  the  plane  ABDL,  or  of  its  trace  AB,  accord- 
ing as  ,'/  =  ^  or  y=0.  Similarly  (2)  is  the  equation  of  the  plane 
AHDC,  or  of  its  trace  AC,  according  as  x=  -^  or  x  =  0.  But 
AB  and  AC  are  the  projections  of  AD  on  XZ  and  YZ,  since 
the  planes  ABDL  and  AHDC  are  perpendicular  res[)ectively 
to  XZ  and  YZ.  Hence  the  straight  line  AD  is  determined 
when  its  projections  on  any  two  coordinate  planes  are  given. 

Eliminating  z  between   (1)   and  (2),  we  have  an  equation  of 

the  form 

A"x  +  B"y  +  F"^0,  (.8) 

which,  in  like  manner,  is  the  equation  of  a  plane  perpendicular 
to  XY,  or  of  its  trace  on  XY,  according  as  we  regard  z  =  ^  or 
z  =  0.     This  plane  is  evidently  the  plane  AOD,  passing  through 


216  ANALYTIC    GEOMETEY. 

the  intersection  AD  of  (1)  and  (2),  since  (1),  (2),  and  (3)  are 
simultaneous  ;  and  its  trace  is  OD,  the  projection  of  AD  on  XY. 
Hence,  in  general,  if  we  assume  any  two  equations  of  the 
first  degree  between  two  variables,  as 

f(x,z)  =  0,    /'(y,  2)=0, 

and  eliminate  the  common  variable,  obtaining  the  third  equation, 

f"{x,y)=0, 

these  three  equations  may  ])e  regarded  as  the  projections  on 
the  coordinate  planes  of  a  straight  line  in  space,  any  two  of 
them  being  sufficient  to  determine  the  line. 

151.  Equations  of  a  straight  line  through  a  given  point  haoing 
a  given  direction.  Let  (x',  ?/',  z')  be  the  given  point,  P',  and 
a,  /?,  y,  the  direction-angles  of  the  given  line.  Let  P  be  any 
other  point  (x,  y,  z)  of  the  line,  and  denote  P'P  by  r.  Then 
the  projections  of  P'P  on  the  axes  are  (Art.  133) 

x  —  x'=reosa,    ?/ —  y'=  rcos/S,    z  —  z'—rcosy, 
from  which  ^~^'  =  y~y'  =  ^^ ,  ( 1 ) 

cos  a  cos/3         cosy 

which  are  the  required  equations,  any  two  being  sufficient  to 
determine  the  line. 

Eq.  (1 )  is  called  the  symmetrical  form. 

152.  To  put  the  equations  of  a  straight  line  under  the  sym- 
metrical form.  » 

The  symmetrical  form  being 

X  —  x' _y  —  y'  _z  —  z' 

cos  a  cos/3         cosy' 

the  condition  that  the  equations  of  a  straight  line  are  in  this 
form  is  that  the   sum  of  the  squares  of  the  denominators  is 

unity.     Let 

X  —  x'  _  y  —  ?/'  _z  —  z' 

L     ~~M~~    N 


THE   STRAIGHT   LINE.  217 

be    the    given    equations.       Dividing    tlie    denominators    by 
Vi/'  +  M-  +N-,  we  have 

•r  —  .>■'  II  —  y'  z  —  z' 


L  "  M  N 


in   wliich  tlie   sum   of    the    squares  of    the   denominators  =  1. 
Thus,  let  3a;  —  22  +  1  =  0,  ix  —  y  =  0,  be  the  given  line.   Then 

X _2z  —  I  _y 
1~       3~"~4' 

Dividing  the  denominators  by  Vl"+3-+ 4^  =  V26, 

a;        2  z  —  1         ?/ 


the  direction-cosines  beins; 


V26        V26        V26 
1  3  4 


"^  V2q'    V26'    V26 

Examples.  1.  Find  the  equations  of  the  intersection  of 
x  —  y  +  z  —  2  =  0^  and  a;  +  ^  +  2  2;  —  1  =  0,  and  determine  the 
position  of  the  line. 

Elimmating  >/  and  z  in  succession,  we  have 

2.T+3c-3  =  0,     .r-3_y-3  =  0, 

X        2  —  1  '/  +  1 

or  -  = =  iLJ— , 

1-2  1 

3  3 

whence  the  line  passes  through  (0,  —1,  1),  and  its  direction-cosines  are 

3 2_        1 

Vu        Vli'    vli 

2.  Find  the  intersection  of  a; -f-^  — 2  +  1=0  and  4a;  +  y  — 
22  +  2  =  0. 

Ans.    A  line  through  (0,  0,  1),   ichose  direction-cosines  are 

1  2  3 

vn'  vTi'  vii 


218  ANALYTIC   GEOMETKY. 

3.  Determine  the  position  of  x=  4z  +  3,  y  =  3z  —  2. 

Ans.   Aline  through  (3,  —2,  0),  irhose  direction-cosines  are 

4  3  1 

V26'    V2g'    V26 

4.  "Write  the  equation  of  a  line  through  (1,  2,  —6),  having 
|,  1,  |,  for  du'ection-cosines. 

Ans.  x-2y  +  3  =  0,  2y-z-10  =  0. 

5.  Write  the  equation  of  a  line  through  (1,  4,  —3)  parallel 
to  Z.  Ans.  a;  =  1 ,  y  —  4:. 

153.     Equations  of  a  straight  line  through  two  given  points. 

Let  {x',  ?/',  2;'),  (a;",  t/",  2;"),  be  the  given  points.     The  equa- 
tions of  a  straight  line  through  {x',  y',  z')  are 


v' 


x  —  x'^y  —  y'^z  —  z  .^. 

cos  a  cos/?  COSy 

Since  the  point  (.t",  ?/",  z")  is  also  on  the  line,  its  coordinates 
must  satisfj"  (1)  ;  hence 

x"  -  x'  _  y"  -  y'  _  z"-z' 


cos  a  COS^  COSy 

Dividing  (1)  by  (2),  member  by  member, 


(2) 


X- 

-x' 

= 

V- 
2/" 

-If 

-y 

= 

z- 

z" 

-z' 

0;" 

-x' 

-2' 

(3) 

which  are  the  required  equations,  any  two  of  which  determine 
the  line. 

Examples.      1.    Write   the    equations   of    the   straight   line 
passing  through  (1,  2,  4),  (—3,  6,  —  1). 

.        x  —\      y  —  2       2  —  4 
-4  4  -5 

2.  Find  the  direction  of  the  line  of  Ex.  1. 

3.  Find  the  points  in  which  the  lino  of  Ex.  1  pierces  the  co- 
ordinate planes.  Ans.   The  line  pierces  XY  in  {— ^.,  ^) . 


THE    STRAIGHT   LINE. 


219 


4.  Write  the  equations  of  lines  through  the  following  points, 
and  find  their  directions  : 

(2,1,  -1),  (-3,  -1,  1);   (6,  2,  4),(-G,  -3,  1). 

5.  A  line  passes  through  (1,  1,  2)  and  the  origin;  find  its 
equations. 

6.  A  line  passes  through  (1,  6,  3,)   (1,  -  6,  2).     Find  the 
equations  of  its  projections  on  the  coordinate  planes. 


154.    To  find  the  angle  behceen  two  given  straight  lines. 
Let 


X  —  x'  _  y  —  y'  __z  —  z' 


N' 


x  —  x"_  y  —  y"  _z  —  z" 


L"  M"  N"  ' 

be  the  given  lines.     The  angle  between  the  two  lines  is  given 
by  the  relation  (Art.  142) 

COS^  =  COS  a' cos  a"  +  COS  y8' COS /3"+ COS  y' COS  y", 

in  which  a',  (3',  y',  and  a",  /3",  y",  are  their  direction-angles. 
But  (Art.  152), 


cos  a'  = 


COS/3'  = 


cosy'  = 


M' 

VL"  +  3/'-  +  N^' 

N' 


(1) 


cos  a"  = 


cos/3": 


cosy    = 


L" 


'   I^ 


M" 

N" 
^L"-  +  M"-'  +  A^"2'  J 


(2) 


220  ANALYTIC   GEOMETllY. 

XT                   ^                  L'V'+M'M'^+N'N"  ... 

Hence      cos^  =  — r===== •         (3) 

Vi''  +  M'-  +  N'-'  VX"-  +  M"'-  +  N"' 

Cor.  1.  If  the  lines  are  parallel,  the  corresponding  direc- 
tion-cosines are  equal,  each  to  each ;  hence  from  (1)  and  (2), 

L'  ^  M'  ^  N' 
L"      M"      N" 
are  the  conditions  of  parallelism. 

CoR.  2.     If  the  lines  are  perpendicular  to  each  other, 
^=90°,    cos^  =  0; 
hence  L'L"+ M'M"+ N'N"  =  0 

is  the  condition  of  perpendicularity . 

Examples.  1.  Find  the  equations  of  the  sides  of  the  tri- 
angle whose  vertices  are  (1,  2,  3),  (3,  2,  1),  (2,  3,  1),  and 
the  angles  of  the  triangle. 

'^''''    I        ^=2J'  .=  ir      2x  +  z  =  hl    C'    2'    3" 

2.  Find  the  angle  between 

?/  =  5.r-f  3,     z  =  ox-\-b^     and    y  =  2x-\-\,  z  =  x. 

3.  Show  that 

4a; -3t/- 10  =  0,     ?/  + 4^;  +  26  =  0, 

and  7.c-2?/  +  26  =  0,     34^  +  7^-90=0, 

are  perpendicular  to  each  other. 

4.  Show  that        x=-2z  +  \,     ?/  =  32!  +  4, 
and  a:  =  ;3-22;,     y  =  z  —  2, 
are  perpendicular  to  each  other. 

5.  Show  that 

2.X'  -  //  +  1  =  0,     3 ?/  -  2 2  +  0  =  0, 
and  2a;  —  ?/  —  7  =  0,     Sr/— 22  +  7  =  0,  are  parallel. 


THE   STRAIGHT   LINE.  221 

6.  Find  the  conditions  that  the  straight  Hne 

x  —  x'_y  —  y'_z—z' 
L     ~     M    ~    N 

is  parallel  or  perpendicular  to  the  plane 

Ax  +  By+Cz  +  F=0. 

The  line  is  parallel  to  the  plane  when  it  is  perpendicular  to  the  perpen- 
dicular on  the  plane.  But  the  direction-cosines  of  the  perpendicular  are 
proportional  to  A,  B,  G  (Art.  146),  and  the  direction-cosines  of  the  line 
are  proportional  to  L,  M,  N  (Art.  152).  Hence  the  condition  of  parallel- 
ism  is  (Art.  154)  ^^^  ^  ^^^^  ^^  ^  0 

The  line  is  perpendicular  to  the  plane  when  it  is  parallel  to  the  perpen- 
dicular on  the  plane;  hence,  the  condition  of  perpendicularity  is 

ABC 

7.  Find  the  equation  of  a  line  through  (  —  2,  3,  5)  perpen- 
dicular to  2x  +  8y  —  z  —  4:  =  0. 

Ans.  a; +  22-8  =  0,  y  +  Sz-43  =  0. 

8.  Show  that  2x  —  y  =  0,  Sy  —  2z  =  0  is  perpendicular  to 

a;  +  2?/  +  32-6=0. 

9.  Show  that  2  =  3,  x-\-y  =  o  is  parallel  to 

.^'-f2/.+  2-6  =  0, 
and  to  the  trace  of  the  latter  on  XY. 


CHAPTER   YI. 

SURFACES   OF  REVOLUTION,    CONIC 
SECTIONS,   AND  HELIX. 


-OOXKOO- 


SECTION  XVIII.  —  SURFACES   OF   REVOLUTION. 


155.  Defs.  A  line  is  said  to  be  revolved  about  a  straight  line 
as  an  axis  when  every  point  of  the  line  describes  a  circle  whose 
centre  is  in  the  axis  and  ivhose  plane  is  perpendicular  to  the  axis. 

The  moving  line  is  called  the  generator,  and  the  surface 
which  it  generates  a  surface  of  revolution.  It  follows  from  the 
definition  of  revolution  that  every  plane  section  of  a  surface  of 
revolution  perpendicular  to  the  axis  is  a  circle,  and  that  every 
plane  section  through  the  axis  is  the  generator  in  some  one  of 
its  positions.  A  plane  through  the  axis  is  called  a  meridian 
plane,  and  the  section  cut  by  such  a  plane  is  called  a  meridian. 

156.  General  equation  of  a  surface  of  revolution. 

Let  the  axis  of  Z  be  the  axis 
of  revolution,  the  generator  a 
plane  curve  whose  initial  posi- 
tion is  in  the  plane  XZ.  and  P 
any  point  of  the  generator. 
Since  the  generator  is  in  the 
plane  XZ,  its  equation  will  be 
X  =f{z),  but  as  it  revolves  about 
Z,  the  .T-coordinate  of  any  point 

FiR.  111. 


'V 


SUEFACES   OF   REVOLUTION.  223 

as  P  will  differ  from  that  of  its  initial  position.  Hence,  to  dis- 
tinguish the  cc-coordinate  of  the  surface  from  that  of  the  gene- 
rator in  its  initial  position,  represent  the  latter  by  r ;  then  the 
equation  of  the  generator  will  he 

r=f{^).  (1) 

But  P  remains  at  the  same  distance,  r,  from  Z  dm'ing  the 
revolution;  hence  (Art.  137) 

r^  =  .r+r.  (2) 

Substituting  in  (2)  the  value  of  r  from  (1), 

^•^  +  ?r  =  [/(^)T,  (3) 

is  the  general  equation  of  a  surface  of  revolution.  In  any  par- 
ticular case  substitute  in  (3)  f{z)  from  the  equation  of  the 
generator. 

157.  The  sphere.  If  a  circle  be  revolved  about  any  one  of 
its  diameters  the  surface  generated  will  be  a  sphere.  Let  the 
diameter  coincide  with  Z  and  the  centre  with  the  origin.  Then 
the  equation  of  the  generator  will  be 

whence  1^  =  \_f{'z)f  =  R- —  z-.  Substituting  this  in  the  general 
equation  x^  -\-y'=  [/(^)]'i  we  have 

o?  +  f  +  z'  =  R\ 
which  is  the  required  equation. 

158.  The  prolate  spheroid,  or  ellipsoid.  This  is  the  surface 
generated  by  the  revolution  of  an  ellipse  about  the  trans- 
verse axis.  Let  the  transverse  axis  coincide  with  Z  and  the 
centre  with  the  origin.     Then  the  equation  of  the  generator  is 

a-r  +  h-z-  =  a^h\ 

whence  t''-=  -^a'-z')  =  [f{z)Y. 

a- 


224  ANALYTIC   GEOMETRY. 

Substituting  this  in  ar  +  ?/-  =  \^f(z)y,  we  have 

cr(^-'  +  /)+&V  =  a^Z>%    or    f^  +  t+^[=i,  (i) 

(r      b-      a- 

If  a^  =  b^=  Br,  the  ellipsoid  becomes  a  sphere.  B}'  definition, 
plane  sections  parallel  to  XY  are  circles.  Let  the  student 
prove  that  plane  sections  parallel  to  XZ  and  YZ  are  ellipses. 

159.  The  oblate  spheroid,  or  ellipsoid.  Tliis  is  the  surface 
generated  by  the  revolution  of  the  ellipse  about  the  conjugate 
axis.  Let  the  conjugate  axis  coincide  with  Z  and  the  centre 
with  the  origin.     Then  the  equation  of  the  generator  is 

whence  r'  =  -'  {b'  -  z')  =  [/(2)]S 

and  &2(.^-  +  2/^)4-aV  =  a^6^    or    t  +  y'^  +  t^l, 

a-      a-      b- 

which  is  the  required  equation.  If  or  =  br  =  R-,  the  ellipsoid 
becomes  a  sphere. 

Let  the  student  determine  the  plane  sections  parallel  to  the 
coordinate  planes. 

160.  The  paraboloid.  This  is  the  surface  generated  by  the 
revolution  of  a  parabola  about  its  axis.  Let  the  vertex  of  the 
parabola  be  at  the  origin,  the  axis  coinciding  with  Z.  Then 
the  equation  of  the  generator  is 

,-'=2p2=[/(^)P, 

and  the  required  equation  is 

x^  +  f=2pz.  (1) 

Let  the  student  show  that  plane  sections  parallel  to  YZ  and 
XZ  are  parabolas. 

161.  The  hyperboloid  of  two  nappes.  If  an  hyperbola  be 
resolved   about   its  transverse   axis,  the  surface   generated  is 


SUKFACES  OF  KEVOLUTION.  225 

called  the  hyperboloid  of  two  nappes.     With  the  centre  at  the 

origin  and  the  transverse  axis  coincident  with  Z,  the  equation 

of  the  generator  is 

a-r-  —  b-z-  —  —  a-b^, 

whence  a-(ar  +  //-)  —  b'-z-  =  —  a^&^,  (1) 

Q  O  9 

X-      v~      z- 
or  -  +  !,--,,  =  - 1, 

0-      b-     a- 

is  the  required  equation. 

Let  the  student  determine  the  plane  sections  parallel  to  the 
coordinate  planes. 

162.  Hyperboloid  of  one  nappe.  This  is  the  surface  gener- 
ated by  the  revolution  of  an  hyperbola  about  its  conjugate  axis. 
Assuming  the  centre  at  the  origin  and  conjugate  axis  coinci- 
dent with  Z,  the  equation  of  the  generator  is 

a-  z-  —  b-  v  =  —  a"  O", 

and  that  of  the  surface  is 

b-{x-  +  f-)-cez''  =  a'b\  (1) 

099 

XT        V-         Z- 

or  — +  ^^,-      =1. 

a-      a-      b- 

Let  the  student  determine  the  sections. 

163.  Cylinder  of  revolution.  If  a  straight  line  revolve  about 
another  to  which  it  is  parallel,  it  will  generate  the  surface  of  a 
circular  cylinder.  Let  Z  be  the  axis  and  r  =  R  the  equation 
of  the  generator  parallel  to  Z  in  the  plane  XZ.     Then 

r^R=f{z), 

and  x'-\-y'  =  Ii'  (1) 

is  the  required  equation,  z  being  indeterminate. 

Let  the  student  show  that  sections  parallel  to  z  are  two  par- 
allel straight  lines,  or  one  straight  line,  elements  of  the  cylinder. 


226 


ANALYTIC    GEOMETRY. 


164.  Cone  of  revolution.  If  a  straight  line  revolves  about 
auotber  straight  line  which  it  intersects,  the  surface  generated 
is  that  of  a  cone.  Any  position  of  the  generator  is  called  an 
element  of  the  cone. 

Let  AB  be  the  generator,  and 
Z  the  axis  of  revolution.  The 
cone  will  be  a  right  cone  whose 
vertex  is  A,  OA  =  h  being  the 
altitude  and  OB  =  R  the  radius 
of  the  base  in  the  plane  X^. 
The  coordinates  of  A  and  B  are 
(0,  h).  (R,  0),  and  the  equation 

h 


of  the  generator  z  = 


R 


+  h, 


whence 


R' 


Fig.  112, 


h- 


,-'  =  [/(.)]^^  =  ^(/i_z)^     (1) 
and  the  equation  of  the  surface  is 

(2) 


or  if  ^  =  angle  which  the  generator  makes  with  X=  angle  made 
by  the  elements  of  the  cone  with  the  plane  of  the  base, 

(x' +y')  tan' e={h-zy.  (3) 

If  the  vertex  A  is  at  an  infinite  distance,  the  cone  becomes 
the  cylinder.     In  this  case  /i  =  x,  and  from  (1) 


uw= 


'^,_2zR-     z-R^ 


h 


h- 


=  R\ 


and  we  obtain  the  equation    of   the  cyUnder   .^"  -\-y^  =  R\   as 
before. 

Let  the  student  prove  that  every  plane  section  parallel  to  Z  is 
an  hyperbola. 


THE   CONIC    SECTIONS.  227 


SECTION  XIX.— THE  CONIC  SECTIONS. 


165.  General  equation.  Let  any  plane  (Fig.  112)  be  passed 
through  the  axis  of  1",  cutting  the  section  LPN from  the  surface 
of  the  cone  and  the  line  LN  from  the  plaue  ZX;  and  let 
XON=^  <^,  the  inclination  of  the  plane  to  XY.  Since  the  cut- 
ting plaue  is  perpendicular  to  ZX,  its  equation  will  be  that  of 

its  trace  LN,  or 

z  =  tan  (f>  •  X. 

To  refer  the  curve  of  intersection  LPN  to  axes  in  its  own 
plane,  let  OF  be  the  axis  of  Y,  and  OX'  =  OX  produced  the 
new  axis  of  X;  then  the  coordinates  of  P  referred  to  the  primi- 
tive axes  are  0M=  x,  3IQ  =  z,  QP  =  y,  and  referred  to  the 
new  axes  are  x'  =  OQ,  y'  =  Ql\     Hence 

y=y\  x=OM=OQ  cos(fi=x' cos(j>,  z=3IQ=0Q  sin<^  =  a7'sin<^. 

If  these  values,  which  are  true  for  the  point  P  common  to 
both  the  plane  and  the  cone,  be  substituted  in  Eq.  (3)  Art.  164, 
we  shall  have  the  equation  of  the  plane  section  referred  to  the 
axes   X'OY.      Making  those  substitutions,   and  omitting   the 

accents, 

(x-  cos-</)  -|-  y'-)tairO  =  {h  —  x  sin  <^)-, 

whence 

2/^  tan-  6  +  .t-(cos-  4>  tan- 0  —  sin- 0)  +  2  hx  sin  (^  —  Jr  =  0, 

or,  since  siu-<^  =  cos-<^  tau-<;i>, 

2/2  tan^ e-\-x^  cos' (/> ( tan' ^  -  ta n-</) )  +  2 7i.i-  sin  0  -  7r  =  0 .    ( 1 ) 

Discussion  of  the  equation.  Being  of  the  second  degree 
Itetween  a;  and  y,  this  equation  represents  a  conic. 

U  <f>>6,  tan-c/)  >  tan-^,  £'-4  AC  (Art.  80)  is  positive,  and 
the  section  is  an  h3'perbola. 


228  ANALYTIC   GEOMETRY. 

If  (f><6,  tan^^<tau-^,  B^  —  iAC  is  negative,  and  the  sec- 
tion is  an  ellipse. 

If  ^  =  ^,  tan-</)  =  tan-^,  B-  —  i  AC  is  zero,  and  the  section  is 
a  parabola. 

Hence  the  section  is  an  h3'perbola,  ellipse,  or  parabola, 
according  as  the  cutting  plane  makes  an  angle  with  the  plane  of 
the  base  greater  than,  less  than,  or  equal  to,  that  which  the 
elements  do. 

If  /i  =  0,  the  equation  becomes 

yHsiu'e  +  x-  cos-<^(tan-6'  -tan^c^)  =  0, 

and  the  plane  passes  through  the  vertex  which  is  at  the  origin. 
In  this  case  if  ^  >  ^,  the  equation  takes  the  form  y=±  ax\  and 
represents  two  straight  lines  through  the  origin.  If  <^  <  ^,  it  is 
satisfied  only  for  x  =  0,  ?/  =  0,  and  represents  a  point.  If  <^  =  ^, 
it  reduces  to  y  =  0,  the  equation  of  X.  These  are  particular 
cases  of  the  hyperbola,  ellipse,  and  parabola,  respectively. 

If  <^  =  0  ;  a  particular  case  of  4>  <0,  the  section  is  a  circle  ])y 
definition. 

If   /i=oc,  the  cone  becomes  a  cylinder.     Putting  h  =  cc  in 

7  2 

Eq.  (1),  after  substituting  for  tan^^  its  value  --  and  dividing 


through  by  h^,  we  have 

y  H-  X-  cos-  (ji  =  li- ; 


ie- 


which  is  the  equation  of  an  ellipse,  except  when  ^  =  0°  and 
(fi  =  90°,  in  which  cases  the  section  is  a  circle,  or  two  paralk'l 
straight  lines,  elements  of  the  cylinder. 

Having  tluis  given  to  ^  all  possible  values  from  0°  to  90°,  and 
h  all  possible  values  from  0  to  infinity,  we  have  found  every 
section  of  the  cone  except  those  parallel  to  the  axis  of  revolu- 
tion, which  latter  have  been  already  considered  in  Arts.  163  and 
164.  Thus  every  plane  section  is  seen  to  be  one  of  the  varieties 
of  the  conies. 


THE   HELIX. 


229 


SECTION  XX.  —  THE  HELIX. 


166.  Defs.  If  a  rectangular  sheet  of  paper  be  rolled  up  into 
a  right  cylinder  with  a  circular  base,  any  straight  line  drawn  on 
the  paper,  not  parallel  to  its  sides,  will  become  a  curve  called 
the  helix.  Or  it  may  be  defined  as  the  curve  assumed  by  the 
hypothenuse  of  a  right-angled  triangle  whose  base  is  tangent 
to  the  base  of  the  cylinder  and  whose  plane  is  perpendicular  to 
the  radius  of  the  base  through  the  point  of  contact,  when  the 
triangle  is  wrapped  around  the  cylinder.  The  helix  forms  the 
edge  of  the  common  screw.  It  follows  from  the  definition  that 
the  helix  makes  a  constant  angle  with  the  elements  of  the  cylin- 
der ;  namely,  the  acute  angle  at  the  base  of  the  triangle. 

167.  Equations  of  the  helix.  Let  the  axis  of  the  cylinder 
coincide  with  Z,  OA  =  li  = 

radius  of  its  base  in  the  plane 
XY,  P  being  any  point  of 
the  helix,  a  =  constant  angle 
at  the  base  of  the  triangle, 
the  vertex  of  this  angle  being 
assumed  on  the  axis  of  X 
at  A,  and  <^  =  ylOQ  =  angle 
made  by  the  projection  of  the 
radius  vector  OP  on  X5^with 
X.     Then 

X  =  0M=  OQ  cos  cfi  =  R  cos  4>^  y  =  ^tQ  —  OQ  sin  <^  =  P  sin  <^, 
z  =  QP  =  base  of  triangle  x  tan  a  =  QA  -  tana  =  P<^  tana. 

Hence,  if  k  =  tana,  the  equations  of  the  helix  are 

a;  =  Pcos<^,  ?/  =  Psin(^,  z='kR<^. 


'» 


'/n^ 


o 


